Not to bury the lede: Is it okay as DM to put pressure on some players to know their character enough not to slow the game unnecessarily by delaying super-rapid advancement from the party.
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So I was planning on starting at 1st and just advancing quickly - leveling every session until 3rd, then almost as fast up to 5th. (This group has been exclusively milestone based levelling under the various DMs, levelling when the DM says so is part of our social contract.)
But I don't want to lose them by going to fast, so I was thinking of advancing as long as everyone seemed to have a good familiarity on the mechanics of their character.
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On reflection though, other players may see that as those players denying them their level up. And if over this much time the players who are slow seem to have indicated through their actions that the mechanics aren't their priority so this may be unfun for them on that side as well. But the goal is to speed up play at the table, which I think is a reasonable goal.
If I were in your game, I would end up among the players who know a lot about their PC, but I would not mind AT ALL a slow advancement. In fact I am most often annoyed by fast advancement
because I want to know my PC abilities well! I can't speak for your players however, but make sure how they all feel about advancement speed, maybe they want to skip low levels but are fine with slowing down around 5th?
For your problematic players, I would definitely try to encourage them to get into enjoying learning the rules better, but obviously you have to find a way to make it sound an encouragement, not a duty, so don't go too far.
OTOH you can really help them with props. Consider at least alternate character sheets, spell/action cards, and tokens/counters.
I run D&D for my children since they were ~7 and I have gradually designed our own stuff, starting with better character sheets. You can find them in the download section but I don't think they are right for you: one of my purposes was to encourage my kids to rely on simple calculations on the fly, so I removed the skills section, but I left the room for precalculating weapons and cantrips attacks (I wanted them to do simple additions every now and them, but not every single round! ). Your case is different but you can figure out better character sheets by eliminating everything that's not useful on a regular basis, try to end up with the smallest possible character sheet for them.
Second, I designed and printed out action cards (spells, special abilities and some common abilities like 2wf), and color-coded them by action type. Having cards in front of you of what you can do on your turn is a lot better than a list on the character sheet, especially when sheets are too often designed for optimising character
creation instead of character
usage in game... see how many sheets organize abilities depending on source (separate sections for racial abilities, class abilities, feats...) instead of when you can use them (combat, exploration, downtime...)? This is the same mistake as organising your tens of thousands of Lego bricks by colour instead of shape: makes it easy to store them the first time, and a nightmare later to find what you need
You can also use colorful counters like poker chips for spells slots or limited resources, to help remember how many they still have.
Just keep in mind to focus on the players in need. These ideas require some work, so don't embark on a quest to design e.g. spell cards for every class and level... just make those you need NOW for those players only otherwise you'll be overwhelmed by work...
