If someone uses Bullet Time, do they have to recycle Run before I can react?

No. Your opponent gets to react before you recycle any runes.

Here's the sequence for Bullet Time:

1. You play it — pay 1 energy, target a battlefield, put it on the chain.
2. Opponent reacts — they can use Reaction cards or abilities while Bullet Time is on the chain, without knowing how much power you'll spend.
3. Bullet Time resolves — now you decide how much power to pay by recycling runes, and deal that much damage.

The power payment happens during resolution, not as an additional cost when casting. So your opponent has to make their reaction decision blind — they can't wait to see how many runes you recycle first. If they pass and you start resolving, there's no reaction window after you've paid.