I live in Bristol.

FRA sector was dropped due to lack of demand in face of competition. Agree that LHR is easily the best option. Heathrow Express is very easy and takes you directly to London Paddington. Fares may be expensive to Bristol if at peak times and not prebooked. Off peak you should be able to walk up and get a cheap fare. Off peak most standard class carriages are fairly empty and you may find First Class too expensive. Short walk to the departures to Bristol. At peak times trains every hour.

Beware there are two main Bristol train stations: Bristol Temple Meads in the city centre and Bristol Parkway located in the northern suburbs and the latter is a good pricey taxi ride into the centre even though the train journey time may look shorter. Take care that you choose the right destination station. The Temple Meads route will take you through the city of Bath and the journey is a lot more scenic.

For a good look at times and fares etc try looking at: http://www.firstgreatwestern.co.uk/

Coach can be a good option with the coaches located within walking distance of the terminals and prices will usually be considerably cheaper than rail. However you will be at the mercy of local traffic conditions on the M4 heading west and if you hit the peak or there is a traffic accident 2 hours can look a little optimistic for the actual journey time.

National Express is the main carrier between LHR and Bristol: http://www.nationalexpress.com/home.aspx

I agree that the Reading Rail Air coach is a waste of time and effort now that the Heathrow Express is up and running. Good luck !!