[RPG] Does the Great Old One warlock’s telepathy from the Awakened Mind feature take an action

I think you're right by RAW; elves' Fey Ancestry and bards' Countercharm only grant advantage on saving throws against being charmed, and therefore can't prevent something inevitable like this. There are a number of spells in the same boat, such as Aura of Purity.

Of course, there's a flip answer of "don't become incapacitated" and "don't let the warlock touch you". The rules are full of ways to do that, although some of them are complicated and most of them are fallible in one way or another. But, let's say that those things are unavoidable...

Prevention

Spells:

• Protection from Evil and Good (1st level cleric, paladin, warlock, and wizard spell) possibly could work, although it only prevents being charmed by aberrations, celestials, elementals, fey, fiends, and undead.
• Magic Circle (3rd level cleric, warlock, and wizard spell) is the same, but curiously leaves out aberrations.
• Hallow (5th level cleric spell) Has the same caveat. This has a 24-hour casting time and consumes 1000gp, but has a duration of "until dispelled", so if you know you're going to be fighting a thrall-creating warlock outsider, this might be a way to prepare.
• Contingency (6th level wizard spell) plus one of the cure spells of 5th level or lower (presumably Remove Curse, since that's the one available to wizards). This isn't quite prevention, since it can't keep the effect from taking hold for an instant before being removed, but I'm going to count it as such, because while it probably feels unpleasant, being charmed isn't that bad when no one has any time for actions.
• Antimagic Field (8th level, cleric and wizard spell) prevents magical charms from affecting targets in the sphere; presumably the "alien magic of your patron" is also temporarily suppressed when thralls are in the field, unless the patron counts as an "artifact or deity".
• Mind Blank (8th level bard and wizard spell) This is probably the winner. Lasts 24 hours (not concentration; whew!) and grants immunity to the charmed condition. It's 8th level, though, so the Create Thrall ability becomes available to players one level earlier.
• Wish (9th level sorcerer and wizard spell) One of the standard effects is immunity to a specific magical effect for 8 hours for up to 10 creatures; you could chose this one. But using Wish in this way risks adverse effects including possibly being unable to ever cast Wish again. If you just need to protect one creature and have access to Wish but not Mind Blank, using Wish to replicate the effect of that spell would be a better option.

Other:

Don't be humanoid. The effect specifically targets humanoids. As I read the rules for effects like Wild Shape and similar, it appears that these make you no longer humanoid, but it also does not seem to be spelled out exactly in the rules we have now. The effects do tend to say "the target's game statistics [...] are replaced by the statistics of the chosen [form]" — but it's kind of nebulous to me whether "creature type" is a game statistic in 5e, or whether affected creatures are "humanoids in beast form" for the purposes of this spell, or truly "beasts". In any case, the 8th-level spell Animal Shapes specifically says that it "turns others into beasts"; if making a DM judgment call, one might separate out the spells that preserve mental ability scores from those like Polymorph, which do not.

Be a high-level berserker. At 6th level, the Path of the Berserker barbarian gets Mindless Rage, which prevents being charmed while raging. A rage ends if the barbarian doesn't keep attacking (or taking damage), which is likely to happen if you're incapacitated. But the 15th level Persistent Rage removes that limitation, so as long as you stay conscious, you're safe — at least, until the 1 minute limit.

Cures

• Monk's Stillness of Mind, which can end a charm effect as an action
• Remove Curse (3rd level cleric, paladin, warlock, and wizard spell) by the ability's own description, of course.
• Dispel Evil and Good (5th level cleric and paladin spell) can end charms with a touch
• Greater Restoration (5th level bard, cleric and druid spell) ditto
• Hallow can also end the charm effect, with the same "outsider" caveat under "Prevention"
• Power Word Heal (9th level bard spell)
• Wish (by emulating Remove Curse)

From Wizards' July 2015 Sage Advice column:

Does the warlock’s Awakened Mind feature allow two-way telepathic communication? The feature is intended to provide one-way communication. The warlock can use the feature to speak telepathically to a creature, but the feature doesn’t give that creature the ability to telepathically reply. In contrast, the telepathy ability that some monsters have (MM, 9) does make two-way communication possible.