I'm late to the party here, but do want to chime in.
Best way from Heathrow to Bath is by coach (bus). Heathrow airport has two coach stations. The coach to Bath picks up from both. We usually arrive at Terminal 5, and walk down the hallway, following the signs that lead you to the coach station. Buy a ticket from the worker in the ticket booth, then wait outside the sliding glass doors for your bus. Easy. The coaches are very nice, with clean, comfy seats (better than most seats on a plane) and WiFi. Large bags are put under the coach in a locked compartment. Carry on bags go with you onboard the coach. You will arrive in Bath at the station that serves as coach station and train station. You can probably walk to your hotel from there unless staying in a remote location.
When you leave Bath by train and go to Moreton-In-Marsh for your Cotswolds visit, your train will pass through Oxford. I advise coming back to Oxford for a day or two after your time in Moreton-In-Marsh. Oxford is lovely, with several excellent museums.
Walking tours by university students (free but please tip) are available at the Tourist Information office.
Just north of Oxford is one of England's most beautiful estates/country houses, Blenheim Palace. You can travel to Blenheim by Oxford city bus. It's a very short ride.
Blenheim is lovely, opulent, and you will see the bedroom where Sir Winston Churchill was born.
This is giving you a house that is a stone's throw from where you will already be, making it easy to get to.
Blenheim was mentioned up this thread by wasleys.
If you take the train north from Oxford, you can reach Warwick (Warwick Castle--beautiful) and Stratford-Upon-Avon.
Stratford-Upon-Avon is a lovely old town. A stay there overnight (or for 2 nights) is needed in order to be there before and after the tourist crowds come in the middle of the day from London for a day trip.
It's easy to return to London from Stratford-Upon-Avon. Take the train to Oxford and onward to London.