Ambushes don't add much complexity, so let's leave the ambush part in there.
The basic rules I use are pretty simple: declare actions in order of Int (lowest to highest) to represent that quicker thinking gives you a shorter OODA loop; all turns occur simultaneously, but actions within a round/turn sometimes need to roll initiative to find out which one goes first; some actions (like Dodge, or maintaining a held action) are considered whole-round activities instead of events within a round, and so they automatically win initiative contests; you can delay your action until everyone else commits to an action, but that makes you automatically lose all initiative contests. (Essentially, you declare Delay as your action, and then you get to declare a new action after everyone else goes.)
So in this case, four heroes are on a cart, and the goblins have all rolled high stealth and won't be detected. The heroes are alert and won't be "surprised", but they do lose initiative automatically (as if they had all implicitly declared Delay, which is the default action).
DM: as you're riding along past a hill past a narrow spot in the road, six arrows suddenly arc in towards you. [Rolls dice] Vlad, you catch a glimpse of a goblin's grinning face in the bushes here right before his arrow hits you for 8 points of damage.
Vlad: can I Shield?
DM: it's only a 14, and I think you would have been alert for possible trouble and aren't surprised, so okay, you Shield. Lose 2 spell points instead of 8 HP. Cranduin, you're hit once too for 4 points of damage; two other arrows clang off your armor. Jack, you got lucky--two arrows were aimed at you but they both missed. There's a brief rustling noise and you lose track of the goblins' whereabouts--they're somewhere within the brush but you're not sure where.
Eladriel (Shadow Monk): guys, let me check this out. I'm hopping out of the cart and making a sweep through the bushes.
Vlad: okay, we'll Delay until she checks it. [Cranduin and Jack nod assent]
DM: El, roll your Wisdom (Perception) check to see if you spot the goblins.
El: 9. [wince]
DM: You don't see anything.
Jack: I'm granting her Bardic Inspiration, and then I'm going to duck down too behind cover and Hide. [starts to roll dice--DM sees it and doesn't stop him because it doesn't look like anyone else is going to declare, and besides the goblins have already gone] 25!
Vlad: I'm going to stop the wagon and crouch down for partial cover behind the edge of the wagon, and Ready a Chill Touch for the first goblin that I see.
Cranduin: I'm going to hop out of the wagon too, to give Vlad some extra cover, and put on my shield and draw my longsword.
DM: Okay, you all do that. Next round. The goblins have all made their action decisions, but since you can't see them I'm not going to tell you what they are, though I suspect you can guess.
Vlad: still holding my Eldritch Blast.
El: Delay.
Cranduin: I'm going to Ready myself to charge over and attack the first goblin who shows his face.
DM: Okay, you'll be ready to attack the first goblin who breaks cover, as long as he is within your 30' movement range.
Jack: I'm still hidden for now, so I'll Delay.
DM: [rolls a handful of dice] Vlad! Three arrows aimed at you--does a 17 hit?
Vlad: Yes, but I'll Shield--oh, stink. I can't if I've already spent my reaction, can I?
DM: Nope. [consults dice, including initiative rolls] One arrow arcs in and misses you, and you blast him right back with Chill Touch. Roll please.
Vlad: 10, miss.
DM: Another arrow misses you, and then a third one, that 17, hits you right in the ribs for 6 points of damage.
Vlad: wait, I forgot about partial cover! My AC this round is 18, not 16!
DM: awesome for you! It hits the wagon right below your ribs.
Vlad: whew!
DM: all three of those goblins fade back into the bushes and you can't spot them any more. Cranduin, what's your initiative this round? The slowest of Vlad's three goblins had a 19 initiative and I doubt you can beat them.
Cranduin: [rolls] Uh, 3.
DM: ...well, I guess you're last. Three goblins also shoot arrows at Eladriel. El, there's one crit, which I assume you're going to try to catch [waits for confirming nod from here] for 11 points of damage minus your missile snatch, and then another 20 which also hits you I think, and then a clear miss.
El: [rolls] I block exactly 11 points of damage.
DM: Okay, you're hit once for 8 points of damage by the second arrow. Cranduin moves to intercept that goblin but he's too slow to hit it before it can try to hide again. However! One of the three that shot at you, the one that got the crit, rolls only a 12 on his Stealth check and you're able to see where he still is and point him out to Cranduin. Go for it, Cran!
Cranduin: [rolls] I got... a 9. Total. I miss.
DM: all right, that still leaves El and Jack with actions for this turn.
El: I attack that goblin, three times including Martial Arts. [rolls] One hit with my staff for 10 points of damage.
DM: And he goes down! Jack?
Jack: Can I very quietly grant inspiration to Cranduin without leaving my hiding place?
DM: Sure. You're like, [whispers furtively] "Fight! Fight! Fight! for the right!" [everyone laughs]
Jack: Okay, I do that.
DM: Okay, round three and you're still facing five goblins, as far as you know. They've got their actions ready but you don't know what they are, and... [etc.]
And that's basically how it works. As you can see, initiative is rolled relatively infrequently*, and the players are as fully-engaged with the game and each other as they would be in a social scene or other noncombat activity. Instead of spending 50-80% of their time sitting around doing nothing, not "allowed" to do anything because it's not "their turn," the players have the freedom to interact with each other and declare actions when they're ready to commit to something, or to wait for a better opportunity later by Delaying. You'll notice that one of the players (Jack's player) is apparently even still thinking more in roleplaying terms ("hide from the monsters!") than in terms of "optimal" tactics like readying attacks or making active perception rolls by Searching.
This style of play should be familiar to anyone who ever read the 2nd edition PHB, since it's almost exactly what AD&D used to use. The main difference is that AD&D didn't explicitly spell out the fact that sometimes initiative rolls don't matter and can be skipped, and it also didn't have the concept of Delaying. (I got the idea of Delay from fencing.)
-Hemlock/Max
* You can see that nothing would change no matter what order the initiative rolls came out in. The only time in the whole scenario when initiative matters is seeing whether Crandruin Readies an action in time to intercept one of the goblins before it can try to Hide again.
This is pretty similar to what I do, although I don't see any point in the delay or the order you establish based on Intelligence. Often in combat you react by reflex or instinct, but regardless, with a no initiative system it's irrelevant.
Here's the same scene with my non-initiative approach:
So in this case, four heroes are on a cart, and the goblins have all rolled a high enough stealth that none of the character's passive Perception overcomes it. Since they are moving at a normal pace and not expecting an attack, and haven't picked up any signs, nobody is making an "active" Perception check.
DM: as you're riding along past a hill past a narrow spot in the road, six arrows suddenly arc in towards you.
The party is surprised. Here's how my system works:
Surprise is usually a Perception check, in this case it was an active Stealth check against passive Perception. But, it could be the case of a Deception check vs Insight (think Kylo Ren and Han Solo).
As a result of the surprise,
-the goblins have advantage on any initiative checks this round (I don't like automatic success rules)
-the goblins have advantage on their first attack roll (which they would have received anyway since they are hidden)
-they benefit from any feature that keys off surprise (such as an assassin).
-the surprised creature cannot use a reaction against the surprising creature's first turn
Note that it's possible for some creatures, on either side of the encounter, to be surprised while others are not.
[Rolls dice with advantage] Vlad, you catch a glimpse of a goblin's grinning face in the bushes
here right before his arrow hits you for 8 points of damage.
Vlad: can I Shield?
DM: No, because he was surprised by the arrows, he cannot use his reaction against them in this turn.
Cranduin, you're hit once too for 4 points of damage; two other arrows clang off your armor.
Jack, you got lucky--two arrows were aimed at you but they both missed.
There's a brief rustling noise and you lose track of the goblins' whereabouts--they're somewhere within the brush but you're not sure where. The goblins can then use their Nimble escape to Hide, which will be against the PCs passive Perception with disadvantage due to the light obscurement.
Eladriel (Shadow Monk): guys, let me check this out. I'm hopping out of the cart and making a sweep through the bushes. Rolls a Wisdom (Perception) check with disadvantage. A 9.
Jack: I'm granting her Bardic Inspiration, and then I'm going to duck down too behind cover and Hide. a 25!
Vlad: I'm going to stop the wagon and crouch down for partial cover behind the edge of the wagon, and Ready a Chill Touch for the first goblin that I see.
Cranduin: I'm going to hop out of the wagon too, to give Vlad some extra cover, and put on my shield and draw my longsword.
DM: Okay, Eladriel jumps out of the cart to start searching through the bushes, meanwhile, Vlad stops the wagon, keeping low for cover, Jack sings a song of inspiration and Cranduin jumps from the wagon with his shield at the ready and sword drawn, Jack then attempts to conceal himself in the wagon, while Vlad begins to utter an arcane incantation.
(End of Round)
Vlad: Still holding my Chill Touch.
Cranduin: I'm going charge over and attack the first goblin who shows his face.
Jack: I'm still hidden for now.
El:Still searching the bushes.
DM:Neither of you have found the goblins yet, what do you want to do?
El:We'll widen our search.
DMWhere? The brush continues in both directions along the ridge, and then woods continue for some distance away from the road.
Cranduin: They probably moved ahead of the caravan to set up another ambush, we'll move that direction.
The goblins actually moved away from the wagon, towards the way they came from and are attempting to go across the road to the other ridge. The goblins make a Dexterity (Stealth) check this round for that action, with disadvantage as they race across the road. Cranduin and Vlad are searching the bushes in the other direction, so their passive Perception is with disadvantage, but the rest of the party uses their passive Perception with advantage against this maneuver. The goblins are successful, however.
(End of round)
You still haven't found the goblins, when suddenly 6 arrows shoot over the ridge on the other side of the cart!
The goblins didn't make an active Stealth check, choosing to fire immediately when in position, so it's their passive Stealth against the passive Perception of the party, but they still have disadvantage due to the goblins being lightly obscured. Regardless, Cranduin was not fooled this time and is not surprised, nor is El or Vlad this time.
DM: [rolls a handful of dice] Vlad! Three arrows coming toward you.
Vlad: Yes, but I'll Shield--oh, stink. I can't if I've already spent my reaction, can I? I'll release my Chill Touch. Rolls dice (10).
Cranduin I run across the road screaming a battle cry at the goblins with my sword high!
El: I race across the road too, to attack!
DM: Vlad, does a 17 hit?
Vlad Yes
DM OK two arrows strike the cart, and then a third one, that 17, hits you right in the ribs for 6 points of damage. The ghostly hand of your Chill Touch lashes out but misses!
Vlad: wait, I forgot about partial cover! My AC this round is 18, not 16!
DM: awesome for you! It hits the wagon right below your ribs.
Vlad: whew!
DM: Cranduin and El are racing across the road, and 3 of them loose arrows at Eladriel. One flies wide, but the other two look on target.
El: I'll try to catch one.
DM: Which one? The first or second.
El:The first. For 11 points of damage.
DM:OK, so as they race across the road, three arrows fly toward El, one is wide, and he catches the next one (lucky, too - that was a critical hit El), and the third hits for 8 points. The goblins are already ducking back into the brush, Cranduin and El, make an initiative check. We need to see if the goblins slipped away before you reached them.
(The slowest of Vlad's three goblins had a 19 initiative)
Cranduin: [rolls] Uh, 3.
DM: Cranduin moves to intercept the goblins but he's too slow to hit it before it can try to hide again. However! (DM rolls Stealth checks for the goblins vs the PCs passive Perception with disadvantage again), El can still see one of them, though, and points it out to Cran, adding advantage to his check, he sees it.
Cranduin: [rolls] I got... a 9. Total. I miss.
DM: all right, that still leaves El's attack, and Jack still hasn't done anything either..
B]Jack:[/B] Can I very quietly grant inspiration to Cranduin without leaving my hiding place?
DM: Sure. You're like, [whispers furtively] "Fight! Fight! Fight! for the right!" [everyone laughs]. But it reveals your hiding place since he must be able to hear you to benefit from it.
Jack: Okay, I do that.
El: I attack that goblin, three times including Martial Arts. [rolls] One hit with my staff for 10 points of damage.
DM: And he goes down!
(End of round)
DM: Okay, round four...[etc.] At this point it is likely that the PCs don't realize that the 6 goblins on the other side of the road were the same goblins that fired on them the first time.
--
Really, what it really comes down to is there isn't really any need for you to add a delay element to the game at all. The actions can either all be declared before the resolution (as in the first round), and the second round.
In the third round, the arrows were a triggering factor and happened first, which determined what the PCs would do in that round. Likewise, running across the road (which we new Cranduin would do since he said he was going to charge the first goblin he saw), El followed right behind, and naturally the other three archers let loose their arrows before they made it across the road. They were arriving just as the goblins would be attempting to hide and get away again, thus the opposed initiative only between the goblins and El and Cranduin.
I also disagree with the assessment of surprise at the start of the combat, alert or not, I think there is always the potential for surprise with an ambush like this. Also, I thought it was a good opportunity to illustrate how I use Stealth, and the fact that I think surprise doesn't have to be only at the start of a combat.
Since they were actively searching a specific area, and looking away from where the goblins were, Cranduin and El were at a particular disadvantage when the goblins snuck across the road. The rest of the group would have been focused on where they expected the goblins to be, but they weren't beating the brush, so to speak, so I didn't give them disadvantage. But they also weren't actively searching so they were passive checks.
