So it struck me that the core fighter abilities cover the basics of what most people want the warlord to do pretty well -- wait! keep reading! -- except for the one issue that they affect the fighter instead of other people. So what if we just, y'know, changed that?
Inspire Second Wind
Once per turn, you can use a bonus action to restore hit points to a friendly creature within 30 feet of you who can hear you. The amount regained equals 1d10 + your warlord level. Once you use this feature, you must finish a short or long rest before you use it again.
Call to Action
Starting at 2nd level, on your turn, you can use your action to allow a friendly creature within 30 feet who can hear you to immediately take an action. Once you use this feature, you must finish a short or long rest before you can use it again.
Coordinate Attack
Beginning at 5th level, once per round, when a friendly creature within 30 feet of you who can hear you takes the Attack action, you can use your reaction to allow that creature to attack one additional time as part of that action.
When you reach 11th level, you gain a second reaction each round and can use this feature twice per round, but only once on the same creature's turn. The number of reactions and uses of this feature increase to three when you reach 20th level.
Indomitable Leader
Beginning at 9th level, when a friendly creature within 30 feet of you who can hear you fails a saving throw, you can allow them to reroll the saving throw. They must use the new roll, and you can't use this feature again until you finish a long rest.
That looks like a pretty good starting point for a 5e warlord to me. Something for both the "a warlord is just a type of fighter" and the "no it isn't!" camps.