RAW, Probably Yes
The contention, of course, boils down to what constitutes a spell being on a given class' spell list, and whether or not that makes it a "X spell". The entry for Celestial Warlock, where its additional spells are listed, has this to say:
The Celestial lets you choose from an expanded list of spells when you learn a warlock spell. The following spells are added to the warlock spell list for you.
Full stop, the class feature specifies that these spells are added to the warlock spell list for you. So for you, as a [part-]Celestial Warlock, the spell Cure Wounds is both a Bard Spell (because of your Bard Levels) and a Warlock Spell (because it has been added to your class list), and therefore, ought to qualify when receiving the Boon of Spell Mastery.
The Boon of Spell Mastery demands that you be capable of casting Cure Wounds (which your level in Celestial Warlock ensures is part of the list), but does not explicitly specify that you be able to cast it as part of that class. It merely says
Choose one 1st-level sorcerer, warlock, or wizard spell that you can cast.
- As a Bard, you have learned Cure Wounds, and are therefore able to cast it.
- As a Celestial Warlock, the spell Cure Wounds is part of the Warlock Spell List for you,
Therefore, you've met the criteria for this Boon, and ought to be able to use it in this manner.
The implication of this is that you probably don't even need to have actual spellcasting levels: the way this is written, A Firbolg Player Character, who will have the ability to cast (once per Short Rest) the spells Detect Magic and Disguise Self, would be eligible for this boon because they are "able to cast" both of those spells, both of which are on the Wizard Spell List, even if they themselves are a Fighter with no spellcasting levels or levels in classes which permit spellcasting.
Note that this might not work for a Divine Soul Sorcerer with Cleric spells, because the specification for them is a little different:
When your Spellcasting feature lets you learn or replace a sorcerer cantrip or a sorcerer spell of 1st level or higher, you can choose the new spell from the cleric spell list or the sorcerer spell list. You must otherwise obey all the restrictions for selecting the spell, and it becomes a sorcerer spell for you.
For Divine Sorcerers, the cause for a spell to count as a sorcerer spell is specifically contingent on having selected to learn it. So in this case, the Sorcerer would only be able to use this boon with a cleric spell if they specifically learn said spell through their Divine Sorcerer features.
Boons are almost exclusively in the realm of DM fiat
While the RAW answer might suggest that this is viable, DMs are not required to provide Boons at all, or at the particular discretion of the players receiving them, or according to the letter of the capabilities specified by said Boons. So if a DM says that they would or would not permit this particular combo, I don't think an appeal to RAW would justify overturning their decision.
Personally, as a DM, I don't agree with the class restriction in the first place (what 1st level spell becomes overpowered by this Boon that wasn't already accessible to those three classes in the first place???), but I can't make assumptions about what your DM will consider reasonable or unreasonable.
There's no such thing as a "Bard version of Cure Wounds"
Some of the other answers to this question have hinged their logic on the assertion that because you learned Cure Wounds as a Bard, that it therefore cannot count as a Warlock spell, irrespective of other features (like the Expanded Spell List feature of the Celestial Warlock Patron); that the spell has an innate quality as a Bard!Cure Wounds distinct from the identity it supposedly needs as a Warlock!Cure Wounds spell in order to be eligible for this boon.
There is no evidence in the 5th edition rules to support this conclusion.
Consider, as a proof-by-contradiction, that we assumed that any prepared spell had this innate quality; that by virtue of being prepared by a Wizard, the spell Detect Magic becomes a Wizard!Detect Magic spell. If this were true, then consider the example of a Spell Scroll, as dictated in the Dungeon Master's Guide:
A spell scroll bears the words of a single spell, written as a mystical cipher. If the spell is on your class's spell list, you can read the scroll and cast its spell without having to provide any of the spell's components. Otherwise, the scroll is unintelligible.
The spell stored in the Spell Scroll would have to have the property of the spellcaster that created it (bear in mind the DMG provides rules for players to create Spell Scrolls), which would mean that a Scroll of Detect Magic would instead be a Scroll of Wizard!Detect Magic if crafted by a Wizard, and as a result, a Cleric would be unable to use it.
This is not how spell scrolls work; they only require that the spell be on your character's spell list. If a Bard created a scroll of Cure Wounds and a Celestial Warlock were to pick it up, they'd be capable of casting it, even without having memorized Cure Wounds; the fact that it was created by a Bard is immaterial.
Therefore, because of how spell scrolls work, we conclude that there's nothing innate about a spell's preparation that makes it uniquely a X!Spell. The only difference is the spellcasting modifier used.
One other wrinkle
It's not clear from context whether the intent of the Boon is to only permit spells that are normally from the Wizard, Warlock, Sorcerer spell lists. It may be that that is the intended interpretation of the wording of this Boon, and as such, it would negate this use of the Boon, but short of a clarifying response from Crawford, we can't make assumptions on this front.