(I briefly thought about offering my shed, but thought better of it. Sorry!)
The stores I know around heathrow ... you might try action bikes
http://www.actionbikes.co.uk/acatalog/stores.html
The staines store is very close to heathrow, and the sheen store would be a reasonable place to get a cab to stop if you were travelling outside peak.
If you want to get your bicycle into central london outside peak hours, you could also use the tube. You can catch a train from heathrow as far as hammersmith, then change to the district line to wherever is handy for your destination.
http://www.tfl.gov.uk/roadusers/cycling/974.aspx
If you want to leave it at heathrow, left luggage is handled by these guys
http://www.excess-baggage.com/airpor...l_services.php