This has been, and will remain, my main actual "dealbreaker"-type concern about 5E. I'm hoping this is just a factor of Clerics being the only real "healing class" in the pregens, and that in practice, any "healing class" will do as well (or close enough as makes no difference). If we're back to 2E levels of "needing a Cleric", though, gosh, that's not going to fly with any of the groups I play with..
If fighters can "heal" in the final product, in any way similar to the ability of a cleric, even in the same ballpark, that will be a dealbreaker for me. Different classes should have their niche. There is no fighter in history or in fantasy who, while having no magical powers, is as good as a modern surgeon, in the midst of the battlefield. This requires magic to be plausible. That sounds funny at first, but it isn't, because without magic healing wounds instantly is impossible, and when a character drops to 0 HP they have received a wound or injury serious enough to knock them unconscious and probably bleeding profusely on the ground, a life-threatening injury. Read the rules, that's how HP are defined, and that's how the game is played, and that's how the game is narrated.
Since it appears that for you, it's a dealbreaker for the game rules to ensure a consistent narrative between various definitions and the result of dice rolls, I have to simply say, I hope your wishes remain unanswered, and if it's a question of your happiness or mine which are at polar odds with each other, I will chose mine.
There is no middle ground here, no compromise. Either non-magical subclasses of fighter are non-magical and therefore do not have access to Cure Wounds or something exactly like it with a different name, or they do, and the game narrative makes no sense, and there is no actual difference between might and magic in D&D. I want there to be a difference, because there in fact is, even in fiction. Certain events and occurrences are logically or physically impossible (stuff that requires retconning events, for instance, like altering the way you narrated a sword chop depending on whether it was a warlord who healed you or a cleric), without magic or the hand of god intervening.
We'll see shortly, I think warlords do not exist in 5e, and can't actually heal others or even themselves. Second Wind was altered to Temp HP, which is really the only inspirational or adrenalin-type HP you need in the game, since it evaporates quickly and doesn't actually heal any prior wounds when you receive them. It only acts as a buffer to prevent incoming damage, which is consistent with the narrative and doesn't require any retconning whatsoever.
I count five classes so far with access to Cure Wounds. That's about half. Isn't that enough? Cure Wounds cures, wounds. That's the source of easy healing in D&D. 5 classes having it is quite easy access. Not all classes should have access to all other classes' powers and magical abilities. That is terri-bad game design in my book.