Not knowing what your budget is, I might recommend that you jump on the Heathrow Express to Paddington and stay somewhere in that neck of the woods if you can't face the onward rail travel (or just want to make a fun night of it). I can recommend
http://www.theroyalpark.com. My MiL stayed there once and loved it, and I can attest that it's lovely as I visited her there.
And there's the Bathurst Deli nearby to grab some croissants and coffee for a picnic in Hyde Park before you rush off to Bristol.
Another (cheaper) option is to just stay at Heathrow - a friend of mine had a 7hr layover on her way to Florence, so she stayed at the
http://www.yotel.com at Heathrow to catch some shut eye. It's in Terminal 4. Interesting idea if you just want to chill after your flight and don't want the hassle of an onward train journey.