According to Jeremy Crawford, yes
Per a tweet from Jeremy Crawford, lead designer of the game, the answer is yes.
The benefit of the Survivor feature happens at the start of each of your turns, whether those turns are inside or outside combat. #DnD - @JeremyECrawford
However, whereas historically his tweets were considered to be official rulings according to Wizards of the Coast, that has since changed, and as a result, such rulings are not definitive.
The rules themselves also strongly suggest yes
According to the Basic Rules section on Casting a Spell, specifically under the subsection for Longer Casting Times:
Longer Casting Times
Certain spells (including spells cast as rituals) require more time to cast: minutes or even hours. When you cast a spell with a casting time longer than a single action or reaction, you must spend your action each turn casting the spell, and you must maintain your concentration while you do so. If your concentration is broken, the spell fails, but you don't expend a spell slot. If you want to try casting the spell again, you must start over.
If turns only existed in combat, spells of longer casting times would not be feasible. Other rules on spellcasting also allude to the same "on your turn" concept for actions, bonus actions, etc, with regards to time to cast.
As a result, any ruling against the Champion healing out of combat would also apply to spellcasting out of combat. While not definitive, it strongly suggests that turns do exist even out of combat.
"...resilience in battle"?
It's been suggested that the use of the words "...resilience in battle" in the feature description implies that it only works in combat. While true that 5e doesn't have flavor text (at least in spells), so all wording should be considered, at the same time there are issues with taking this to imply that all such wording must have applicable and literal meaning.
First, "battle" is a broad and vague term that doesn't find real definition in 5e. "Combat" is well defined and described, but "battle" could just as well mean a prolonged conflict between armies. Without a proper definition of what this means, we can't properly narrow down its application.
Second, there are other examples where such summary descriptions are clearly not intended to be taken literally. Otherwise, Rogues could only use Sneak Attack on "enemies", only when they attack "subtly", and the enemy must be "distracted" (not to mention that "Sneak Attack" is not literally limited to attacks while sneaking). Likewise, a Life Domain Cleric's Channel Divinity: Preserve Life feature would only heal creatures that are "badly injured", not just moderately injured. If the use of the phrase "in battle" is taken literally to that degree, so too must these.
Lastly, we also have specific counter-examples of what such "only in combat" restrictions look like. Consider that a Rogue's Cunning Action ability states: "You can take a bonus action on each of your turns in combat". Cunning Action specifies turns in combat, whereas Survivor does not. If the intent was only for combat turns, this should have been included here.
Adding to the designer intent affirming that this specific feature is intended to be used in and out of combat, there's a solid case that "in battle" is effectively irrelevant.
