Updated : 21 May 2013So far I know of 3 places where you can recycle your old bike or buy a recycled bike in Coventry:
Coventry recycled cycles
The Re-Cycle project - Coventry Jesus Centre
I have also seen nice looking bikes on the back of the scrap metal trucks several times. They might be willing to sell it to you for a couple of quid if you are into restoring.
Look at the Freecycle Coventry site. I have noticed people giving away old bikes a couple of times. You will need to sign up. I once got a really nice mountain bike through the site, while I was studying at Warwick Univ.