Buses in Bath and at Heathrow?

Around Heathrow- the answer is in two parts-
if you are using a Red London bus then no cash is accepted- an Oyster Card or contactless method is required; if you are using a Country service (not a Red London Bus)- for instance to Windsor- then you can pay cash.

For Bath- there isn't a travelcard, and cash is accepted. But, depending how long you are there, if you do more than 2 or 3 journeys a day you will find it more advantageous to buy a day, 3 day, 5 day or week ticket- either as a cash ticket from the driver- or by tap on tap off contactless capping- see this guide- https://www.firstbus.co.uk/sites/default/files/public/node_images/J13198%20July%2023%20Ticket%20Guide_0.pdf

The capping also works for periods not covered by the paper rover tickets- 2 day, 4 day and 6 day.

It is unlikely that you will be better off paying the £2 capped cash fares for each trip- but you need to do the maths for your own itinerary.

The only thing I'll add is....you can travel from Heathrow to Bath by bus (coach) and from Bath back to Heathrow by bus (coach).

There are no trains that go directly from Heathrow to Bath, or from Bath to Heathrow.