Heathrow to the Cotswolds

Moreton-in-Marsh is a short train trip from Oxford, if that helps. I actually stayed in Oxford and took a van-tour of the Cotswolds that departed from Moreton-in-Marsh.

Now we need for someone to tell us the best way to get to Oxford from Heathrow. (From Gatwick, I think it's by bus.)

Coach from LHR to Oxford

https://www.nationalexpress.com/en

Train from Oxford to MiM

http://www.nationalrail.co.uk/

http://www.traveline.info/

From Heathrow take a taxi to Slough Train Station (8 miles), then the train to Moreton-in-Marsh (it’s an hour and twenty minute train ride).

There is quite a regular bus service from Heathrow T5 to Slough.

Another alternative is the Railair Coach to Reading, from both T2/3 and T5.

Personally I wouldn't take a coach to Oxford because you would have to walk between Gloucester Green and the Railway Station.