Who's on first: Oxford or Bath?

The biggest advantage to starting in Oxford is that you can take The Airline, an express bus that leaves directly from Heathrow and goes straight to Oxford. So no need to transfer through London. The bus costs £25, is very comfortable with good room for luggage and gets you there in about 90 minutes. It leaves from Terminal 5 and also the Central Bus Station (near Terminals 2 and 3). https://www.theairlineoxford.co.uk/oxford-to-heathrow-bus/

ETA: I forgot to say that you can book in advance or just tap-and-pay when you board, which is an advantage at the airport in case your flight is delayed. Same price either way.

Thanks, Mardee. This is helpful!

I second the bus to Oxford; I have taken it a few times while visiting friends in Oxford and even though it has been a few years, it never failed me!

The basic problem with the National Express [NE] bus from Heathrow to Bath (and it's competitor, Flixbus, who run a few minutes before each service) is it's lack of frequency- on a Monday afternoon 'later afternoon' direct departures are at 1510, 1820, 2120 and 2315.
You will find a fair few other NE services where you go to Bristol, then the through ticket they sell you is on an all stops service bus back to Bath (that is very unclear on the NE website). Even if you wanted to do that don't buy a through ticket as you are way over-charged for the privilege. Buy an NE ticket to Bristol then a separate £3 single ticket on the service bus.
The Bristol to Bath service bus (#X39) takes at least an hour, but runs every 20 minutes.

Another vote for the Oxford Bus Airline. It's a good idea to look at the stops in Oxford in advance so you can select the one most convenient for you. Anita