Bus or train from Bath?
Probably best to use the bus as if using the train, you would have to change to a bus in any case at Bristol Temple Meads as the airport does not have a rail connection.https://www.airdecker.com
In our opinion, neither is a good way just because of the hassle factor. It's really not too long a distance to rule out a private car service. It will cost a little more, but we think it is well worth it.
We used a private car service based in Bath to go from there to the Bristol airport (and it's much closer than Heathrow) and were very pleased. I don't remember the cost, but it was reasonable. The service we used was Chauffeur Travel, and the web site is https://chauffeurtravel.co.uk/
Send them an email explaining your needs and ask for a quote. Can't hurt to ask.
I would also recommend a taxi. There is a bus service but it will take a lot longer it’s not that frequent or cheap. The train station in Bristol is nowhere near the airport. Bristol airport is very poorly connected by any transport method.