Roughly, what percentage of arrows can be reused after a battle?

I seriously doubt anyone ever kept records of that. In the heat of battle you're too busy to make notes, and afterwards it just doesn't matter unless you started doing it for the sole reason of there being none of your own arrows left and you wrote that in your memoirs, which would be unlikely for the above mentioned reason to have any detailed numbers, only mention of the fact.

It's certainly not inconceivable that it would happen, especially during sieges.
But do keep in mind that it would require reasonably similar bows between the different armies.
If the English were using longbows requiring 1 meter long arrows and the French were using crossbows using 30cm long bolts for example, there'd be no way to reuse those (though a Frenchman might in an emergency be able to break a recovered English arrow in pieces and shoot off those, without fletching it would be highly inaccurate).
THAT makes it unlikely in the case of especially the mentioned English longbow which was a pretty specific weapon not used AFAIK by anyone else.

The comment is made elsewhere that archers didn't collect arrows during a battle:

I agree with Felix Goldberg, the archers most probably didn't reuse the arrows in a single - unbroken - battle, since they had to keep formation and received orders. They might pick up the arrows only if it is there, pinned into the dirt and healthy (like they were targeted by enemy archers), but this is unlikely too, archers typically used against footmen and cavalry.

However, this source for the Battle of Crecy explicitly states (my emphasis)

Each successive charge was weaker and during brief pauses in the battle, the English archers stood in their lines with remarkable discipline, only going down the slope far enough to collect their arrows.

I have seen this and similar comments made elsewhere, though this is the only source I can locate just now. It is important to remember that each of the reported charges by the French knights only lasted a few minutes, say 5 or 6 at the outside, as any charge lasting longer has lost it's most important advantages, speed and momentum.

Each of these charges would have required a much longer time period, perhaps 20 or 30 minutes, for the participants to rally, form up in units, and move to their respective start zones. During these periods there was ample time for designated individuals to run forward and collect arrows.

A quick look at the attached map of the battlefield for Crecy makes clear that the launch zone for the French knights was located 3 or 4 times effective arrow range from the English lines, so the question of safety for archers running forward really doesn't exist.

My understanding is also that the archers and runners running forward to collect arrows had knives and daggers which could also be used to kill any wounded enemies who attempted to resist such endeavours.

So while a specific value for the percentage of arrows that can be reused during a battle is unavailable, some simple calculations regarding maximum troop coverage of ground during a charge (at the trot/canter and then gallop, coupled with the inherent inaccuracy of bows used essentially as artillery, suggests that perhaps 90% of arrows fired fell harmlessly to the ground (or deflected with minimal damage from armour)and that most of these could be reused as soon as collected.

Update

Note also that the arrowheads of broken arrows are themselves valuable, even if not immediately reusable. I know of no direct claim or evidence that spare arrow shafts and fletching were carried in addition to the arrow supply; but not doing so would seem gross incompetence for an army reliant on its archers.

I decided to make an answer since I pointed out a lot of the issue, I deleted my comments, and I state now I don't have the historical evidence, but have a good view on practical use of arrow and bows.

The arrow rarely breaks in the middle, most of the cases if it hits solid material, it breaks very close to or at the head. Both arrow's body and head is recoverable if it found.

I agree with Felix Goldberg, the archers most probably didn't reuse the arrows in a single - unbroken - battle, since they had to keep formation and received orders. They might pick up the arrows only if it is there, pinned into the dirt and healthy (like they were targeted by enemy archers), but this is unlikely too, archers typically used against footmen and cavalry.

Quote: "However, if I'm shooting arrows at guys wearing armor, then some of the ordinance will get damaged after use and not be reusable in "tomorrow's" battle."
This is wrong, they can be repaired very easily. For me in a workshop it takes 5 minutes, with a proper tools and practice in the medieval age it would take similar length of time, it needs to glue together, which happens overnight. And I can tell, it is way easier and quicker just to repair your arrows than making new ones. A typical arrow is reusable 5-6 times until it gets shorter. If it gets too short: you need to make new body. If it is not sharp enough, just sharpen it. Sharpening with proper tools happens in minutes.

I also want to point out, that there are no really huge collection of arrow heads in archeology, so it seems they were reused.

My point would need confirmation from a person who really researches the battle histories. But I would assume the recycle of arrows were lot closer to 100% than 0% for the winner side.