What turn rates are expected by ATC when given expedited turn instructions?
The air traffic controller expects you to 1) start your turn immediately and 2) turn as quickly as can safely be done. We do not have all the facts, and do not know exactly what sort of turn you are capable of doing. Using a phrase such as "expedite turn" or "turn immediately" basically means - I've screwed up, you're in danger, watch out and get out of the way. It is then up to you as pilot in command to avoid a collision, by flying your aircraft and knowing its limitations.

When ATC separation has been lost, you should respond promptly to any instructions issued with the purpose of re-establishing separation. We will also, as far as practical, issue traffic information, and expect you to look out the window and take any avoiding action you find appropriate to prevent a collision.

Finally, as a side note, if you have a TCAS, please remember that a TCAS RA takes precedence over any instruction issued by ATC.