Our Bard had the Feeblemind spell cast on him and failed his save. The battle ended and we gained a level. Feeblemind states that:

On a failed save, the creature's Intelligence and Charisma scores become 1.

While under the effects of Feeblemind, would an ASI be permanently wasted if taken in Intelligence or Charisma?

We use milestone leveling, and it happens immediately once the DM says we've leveled.

The DM shouldn't do this

RAW seems to allow your interpretation.

However, this is not how it's intended to work

ASIs are class features, same as a Rogue's Sneak Attack or a Barbarian's Rage.

There is no precedent for spells or abilities that are able to permanently deprive a player from a class feature. It would have a huge impact and would definitely deserve its own clause in the spell text, were it the case.

It's a good way to alienate your players

ASIs are powerful features and are expected by the players, since class features are usually guaranteed upon leveling. Denying them an ASI just because they failed a save might be enough to put them off the game (especially if they are the min-maxing type).

Unclear...

...but a fair DM would allow the normal score to be increased, although the poor Bard won't get the benefit until the Feeblemind effects have been lifted. Otherwise the DM would basically be causing the character to essentially lose a one-chance-only benefit.

The Feeblemind spell is rather ambiguous as it has an instantaneous duration that causes a lasting effect, and yet the spell also talks about how the target can save or receive specific restorative magic to "end the spell".

Slight contradiction in terms there!

Yet the intention is clearly that once the save has been made or an appropriate spell received then the target's ability scores return to the value they had before. And a reasonable DM would allow this original ability score to be the one that is increased by the ASI.

My interpretation...

I think of it as the same way as an instantaneous damaging spell causes lasting HP damage, but does not (normally) cause your max HP to be reduced. Resting or healing restores HP back to your maximum.

Think of your normal ability scores as the 'maximum' values. Feeblemind damages your Intelligence and Charisma, reducing them to 1, but has no effect on your maximum values. When the "spell ends" (or, rather, you make the save or receive restoration magic), that ability score damage is healed and your scores go back to their original 'maximum' values.

It is not wasted.

Feeblemind is a spell effect. It sets the ability scores to 1 while it is in effect. However, the creature's original scores are not erased and the ASI is applied to these original scores. Take note that under "Specific beats General" the original ability scores are too general , under the spell's specific directions. You have two ability scores for INT and CHA during the effect of feeblemind, but the more specific wins. It is like a Barbarian wearing armor. He has two sources of AC, but can only choose one. In this case, you can't choose but there is a precedent for two values for one trait. When the spell ends, the original scores (modified or not by ASI) return.

On a failed save, the creature’s Intelligence and Charisma scores become 1.

This effect surpasses the ASI. Do not apply the ASI to the score of 1.

At the end of every 30 days, the creature can repeat its saving throw against this spell. If it succeeds on its saving throw, the spell ends.

The spell can also be ended by greater restoration, heal, or wish.

When the spell ends, the effects cease to be. The INT and CHA scores return to their original values. If any ASI was applied to them, they take effect immediately.

When you reach 4th level, and again at 8th, 12th, 16th, and 19th level, you can increase one ability score of your choice by 2, or you can increase two ability scores of your choice by 1. As normal, you can't increase an ability score above 20 using this feature.

However, your DM may rule you cannot apply the ASI to either score during the effects of the feeblemind. The wording "can" implies player agency. This DM sees no problem in allowing a player to postpone applying the ASI to a later date (they are losing the improvement during that time anyway).