The Bristol Bus Station is not far from the Bristol Temple Meads train station. We made the trip on May 10 after an overnight flight arriving Heathrow at 6:30 am. I booked our train to Bristol Temple Meads for after 11. We got thru landing & Passport Control in record time plus we only had carry on bags. I think we were at Paddington Station via Heathrow Express around 9 or earlier. But at 6:30 am -- the morning may have been a bit slower for Passport Control than a 9:55 arrival. I have to agree an 11 am bus departure is cutting it close, if you have luggage to pick up and if you are delayed due to the number of people going through Passport Control.
Once at Paddington we purchased our Two Together Railcard for an advance purchase train ticket to Bristol Temple Meads that I purchased on the website in March. We also had time to grab some food at Paddington. Our problem was we were so fatigued we decided to ante up & still use our Two Together Railcard & bought tickets for an earlier train to Bristol--in our case a good decision since I had allowed way too much time for all the what ifs on our arrival day. (The cards require passport size photos which we had made in a photo booth there.) I initially looked at a bus versus a train and prices seemed reasonable if purchased in advance for both. BUT for us the deciding factor (barring railway work) was that buses travel on roads and highways and are subject to being stuck in traffic like cars, but trains have a dedicated travel network of rails. We thought it would be faster and not only in theory, a bus/coach trip to Bristol from Heathrow was longer than a train ride from Paddington.
Good luck, Sonja
Coach travel for longer distances than bus travel is also quite nice.