When do I have to tap my chem-baroness legend after holding?

You tap immediately — as the "When you hold" trigger is finalized on the Chain, right after the Hold resolves, before the Gold token is created and before your opponent gets any window to respond.

Chem-Baroness's "you may exhaust me to play a Gold gear token" is a triggered ability whose cost follows the opening "you may," so the exhaust is its base cost (Rule 383.3.b). That means the choice and the tap happen at finalization, not on resolution: you decide then whether to pay, and if you do, she is exhausted before the ability even becomes a Finalized Chain Item (Rule 383.3.b.1). Finalizing doesn't pass priority (Rule 337.1.a), so there's no chance to wait, see what your opponent does, or tap "later in the turn" — if you don't tap at that point, the trigger is removed and you get no token.

One consequence: if Chem-Baroness is already exhausted when the trigger finalizes (e.g., you've already used her this turn), you can't pay the cost (Rule 414.4) — the trigger stays, but you must decline and you get no Gold.