You could also get the coach from LHR to Reading station, and get on the train to Bristol there (the trains leave London Paddington, then call at Reading about 30 minutes later, then go on to Bristol). The buses from LHR to Reading go about every 20 minutes and take about 45 minutes. See http://www.railair.com/

It's more cost effective than taking the Heathrow Express into Paddington, then the train from there, but the HEX will probably be more comfortable. Time wise both routes will probably be about the same door to door I guess, though if you were lucky with bus times, it might be quicker to go to Reading

The other advantage if you go into Paddington is you board the train where it starts, so getting a seat is normally not an issue.