Train tickets London to Moreton-in-Marsh, Bath

You can easily buy an Anytime ticket at Paddington to Moreton in Marsh for £42.10. These tickets never sell out and buying at the station is the same price as buying online.

Alternatively, an Advance ticket must be pre-booked and currently costs £14.60 if you are prepared to commit to a particular train. You can buy tickets from this operator https://www.lner.co.uk/ as sometimes https://www.gwr.com/ doesn't process US credit cards.

From the airport to Bath may be quicker by coach (2-hour estimate) and certainly less complicated.
https://www.nationalexpress.com/en/destinations/bath/heathrow-to-bath

if you don't want to commit yourself to an Advance Single the cheapest tickets bought on the day would be an Off Peak Single at £26.90 to Oxford and a Cotswold Discovery ticket covering Oxford to MIM at £10.50
You can buy these at Paddington and you stay on the same train
This should also cover any onward journey by bus if you were travelling on to somewhere like Chipping Campden. I say should as bus route will have three different operators in the last few months.

Note that most weekends in September that the train from MiM will not go through Oxford due to engineering work. You would need to take the substitute bus from Hanborough to Didcot to pick up the train to Bath.