(Note: This answer was written prior to the December 2021 erratum. I still believe that this is the correct interpretation for the general case of class features gained through means other than class levels, and that the erratum represents a specific exception to the general rule.)
Spells from warlock class features use your Charisma
In the class description for the warlock (as with every other spellcasting class), there is only one definition of your spellcasting ability:
Spellcasting Ability
Charisma is your spellcasting ability for your warlock spells, so you use your Charisma whenever a spell refers to your spellcasting ability.
Many Eldritch Invocations grant you the ability to cast spells without telling you which spellcasting ability to use. However, any time a warlock class feature references your spellcasting ability (even indirectly through the description of a spell granted by that feature), it must be referring to your Charisma score, because there is no other spellcasting ability it could possibly be referring to. Having levels in one or more other classes doesn't change this, as explained in the rules for multiclassing:
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell.
The Eldritch Adept feat says (emphasis added):
[...] you learn one Eldritch Invocation option of your choice from the warlock class.
This is a bit redundant, of course, because as far as I know there are no Eldritch Invocations not associated with the warlock class, but it makes the class association explicit: you are gaining a feature specifically associated with the warlock class, which means that the relevant spellcasting ability must be the one defined in the warlock class: Charisma.
In your case, since your bard spell save DC is 12, then your warlock spell save DC is probably also 12, since they both use Charisma as their spellcasting ability. But note that you are not using your bard spell save DC; you are just using another spell save DC that happens to be the same. For example, if you attuned a Reveler’s Concertina, the spell save DC of your bard spells would become 14, but your warlock spell save DC would remain at 12. Alternatively, even if you had no bard levels at all (and hence no bard spell save DC), your warlock spell save DC would still be 12.