I played a little bit of Warlord in 4E, and think you can still capture some of the unique versatility of the class in 5E with the Fighter class and several feats.
I'm fiddling with a build right now that by level 10 I'll be able to do the following basically at-will:
1) Reaction
- When a creature you can see attacks a target other than you that is within 5 feet of you, you can use your reaction to impose disadvantage on the attack roll.
- If you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage, you can use your reaction to take no damage if you succeed on the saving throw
2) Bonus Action
- You can take a bonus action on each of your turns in combat. This action can be used only to take the Dash, Disengage, Hide action, Dexterity (Sleight of Hand) check, use your thieves’ tools to disarm a trap or open a lock, or take the Use an Object action (i.e. Healing Kit to tend to a creature and restore 1d6 + 4 hit points to it, plus additional hit points equal to the creature’s maximum number of Hit Dice. The creature can’t regain hit points from this feat again until it finishes a short or long rest.).
- If you take the Attack action on your turn, you can use a bonus action to try to shove a creature within 5 feet of you with your shield, either to knock it prone or push it away from you.
3a) Use Action to Support
- You touch one willing creature. Once before the spell ends, the target can roll a d4 and add the number rolled to one saving throw of its choice. It can roll the die before or after making the saving throw. The spell then ends.
- You touch one willing creature. Once before the spell ends, the target can roll a d4 and add the number rolled to one ability check of its choice. It can roll the die before or after making the ability check. The spell then ends.
- Use the Help action granting advantage
3b) Use Action to Damage
- Two attacks, ranged or melee: +7 to hit and 1d6(javelin) + 3 + 2d6(sneak attack)
- Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you’re able to make multiple attacks with the Attack action, this attack replaces one of them.
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That's every round. One round you might give an enemy disadvantage, knock another enemy prone, stab it with a javelin with advantage and sneak attack damage, then push it away. The next round you might give an enemy disadvantage, heal an ally, and toss two javelins at that mage in the back. I'm playing a Fighter1/WarlockX build in another game and am surprised by how much mileage I get out of Protection. It is really, really strong.
This doesn't include all the stuff this build can do after rests: Maneuvers, one casting of Bless, Inspiring Leader, Action Surge. To me, this feels very much like a Warlord. If someone gets a nasty spell lingering on them, toss them a bonus to saves via Resistance, then Action Surge, then use the Help action or cast Bless. Clerics can't do that.
Anyway, here is my build for a Warlord-y Fighter:
15 STR, 13 CHR, 13 DEX, 12 CON, 10 INT, 10 WIS
Human - Variant rule, +1STR, +1CHA, Bonus Feat (Healer)
Hermit (Medicine, Religion)
L1 - Fighter1 - Protection, Second Wind, Skills (Athletics, Intimidation)
L2 - Fighter2 - Action Surge
L3 - Fighter3 - Battle Master (Rally, Feinting Attack, ?)
L4 - Fighter4 - Feat (Shield Master)
L5 - Fighter5 - Extra Attack
L6 - Fighter6 - Feat (Magic Initiate - Resistance, Guidance, Bless)
L7 - Rogue1 - Sneak Attack (1d6)
L8 - Rogue2 - Cunning Action
L9 - Rogue3 - Roguish Archetype (Thief - Fast Hands), Sneak Attack (2d6)
L10 - Rogue4 - Feat (Inspiring Leader)
L11 - Fighter7 - Know Your Enemy, +2 Maneuvers
L12 - Fighter8 - +2 STR
L13 - Fighter9 - Indomitable
L14 - Fighter10 - d10 Superiority dice, +2 Maneuvers
L15 - Fighter11 - Extra Attack
L16 - Fighter12 - +2 STR
L17 - Fighter13 - Indomitable (two uses)
L18 - Fighter14 - Feat (?)
L19 - Fighter15 - Relentless (1 die at start if out)
L20 - Fighter16 - Feat (?)