I inquired at the National Rail Express as to their schedule to Bath and read it would be over 4 hours. Any other suggestions? Contacted Mad Dog Tours and their service would be $260.00.
The URL below will take you to a prior discussion of this subject.
http://www.ricksteves.com/graffiti/helpline/index.cfm?topic=3452
If you haven't already made your plane reservations, you may want to consider flying into Bristol instead of Heathrow. Bath is only a few minutes away by rail, and Continental has non-stop service from Newark Airport.
Check NationalExpress.com - they have coaches that leave Heathrow every 90 minutes or so - it's about a 2 hr. 35 min. trip. It's about £17 one-way ($35).
Another option would be to take a bus to Reading. There is a coach service (RailAir.com) that runs pretty frequently (2 or 3 times an hour, I think) from Heathrow to Reading. It takes about 45 minutes to get Reading from Heathrow. From Reading, there are frequent trains that stop in Bath. The train trip from Reading to Bath is about 1 hour.
To guarantee a seat on a specific bus, buy your ticket and make a reservation online before you go. I did this for an Oct. trip and it worked fine. It cost about $18. The bus was full. Allow LOTS of time between landing and catching your bus. Passport control etc can take a long time. My plane landed at Heathrow at about 7:30 am and I arrived at the bus station at 9 am. My ticket for Bath was for 10:15 am. Cut it too close and you may miss your bus and have to wait several hours for one with an available reservation.
Take Heathrow Express to Paddington station (15 min), catch train to Bath (less than 2 hrs.).
I agree with the previous poster that suggested National Express Coaches. That's the least expensive option and only takes about two hours from London.
The NE Coach that departs London Victoria Coach station also stops at Heathrow on the way to Bath. I travelled that route in September, and it works well.
Happy travels!