The College of Glamour bard's Mantle of Inspiration feature (XGtE, p. 14) is worded this way:

As a bonus action, you can expend one use of your Bardic Inspiration to grant yourself a wondrous appearance. When you do so, choose a number of creatures you can see and who can see you within 60 feet of you, up to a number equal to your Charisma modifier (minimum of one). Each of them gains 5 temporary hit points. When a creature gains these temporary hit points, it can immediately use its reaction to move up to its speed, without provoking opportunity attacks.

If you already have temporary hit points when you would gain these new temporary hit points (and you had more before, thus not gaining any from this ability, because temporary hit points don’t stack), would you still be able to use your reaction to move up to your speed?

My understanding would be that no, you couldn’t because if you could, the ability should have been written like this:

When a creature would gain these temporary hit points, it can use its reaction to move up to its speed, without provoking opportunity attacks.

For example, if you have 15 temporary hit points and a bard uses this ability to give people temporary hit points (let’s say 8). Since you already have more temporary hit points than the ability would give you (and temporary hit points don’t stack). Can you still use your reaction to move to your speed, even though you couldn’t accept those temporary hit points?

That is, does it count as you gaining them anyway (even though your total temporary hit points didn’t change, allowing you to use the reaction)? Or does it count as you not gaining temporary hit points (because you already had more than what the ability could give you at the time, and thus not allowing you to use the reaction to move up to your speed)?

It could work even with previous hit points

The critical thing here is how temporary hit points stack, or don't stack specifically:

Healing can't restore temporary hit points, and they
can't be added together. If you have temporary hit points
and receive more of them, you decide whether to keep
the ones you have or to gain the new ones. For example,
if a spell grants you 12 temporary hit points when you
already have 10, you can have 12 or 10, not 22.

--- Player's Handbook, p. 198

So, lets follow an example through: A creature has a number of temp. hit points, let's say 7 and then is affected by Mantle of Inspiration. They can now choose to keep their 7 or gain the 5 instead. If they choose to keep their old, they haven't gained temp. hit points from Mantle of Inspiration and so aren't eligible to use their reaction. But they can choose to gain the new (and fewer) temp. hit points and so be able to use their reaction.