Short and sweet.

Did the first generation mini run from 2001 to 2006?, thus makes me think a second hand exhaust from a 54 plate mini one, will fit our 52 plate mini one.
Is this correct?.

Brownskins

Not necessarily ... changes are sometimes made to parts between model years and even during the same model year.

Normally I tell members to check part numbers on RealOEM. In this case it's kind of confusing ...

RealOEM lists two different part numbers:
RealOEM - R50 One Rear Exhaust System (http://www.realoem.com/bmw/showparts.do?model=RA32&mospid=48013&btnr=18_0476&hg=18&fg=20)
If you use an early build date to find these part numbers it says the parts were for cars built prior to April 2004, yet if you use a build date later than this it shows no "Up To" date.

I'd check with the parts department at your local dealer.