There are many GP's listed in Coventry that seem to do Private as well as NHS.
however,I don't think you would be very comfortable with a GP who was housed in an NHS surgery?
BUPA have a Private GP Service in their Solihull Wellness Centre.
The advantage of this is,if you need ongoing Private Treatment then it can be arranged with BUPA(you don't have to be a member).
This is the BUPA GP Services page:~
http://www.bupa.co.uk...reatments/gp-services
and this the actual webpage of The Solihull Centre:~
https://www.bupa.co.u...sp/solihull/index.jsp
From the map I looked at Solihull doesn't seem too far from Coventry?(I hope)
Good Luck.