Hi Dunbar ,
This wikipedia link:
British_Airways_Flight_9
describes something like your question. I think the crew shut the first engine down (No 4) using the the "Fire Engine Check List" and pulled the Fire Control Handle (Equivalent to Fire Push Button). After they had lost all 4 engines, the crew decided to attempt to relight them. The first engine to be relit successfully was - No 4. The crew remembered to restore the FCH to it's normal position even though it wasn't mentioned in the QRH then.
A fantastic job!
Provided the Fire Push Button is back in it's normal position, then the engine can be restarted - the only difference is you'll have to reset the generator with an off / on selection when ecam asks you to.