Heathrow to Oxford transport

No, but a recommendation for a coach service: https://www.theairlineoxford.co.uk

The coach (bus) service Badger linked to worked very well for us, with a couple of stops in central Oxford. We didn't use it returning to the airport.

We took the bus that Badger recommends. It was very easy.

If you're staying in Oxford, ask them for a recommended car service if that's what you prefer. I've shared a taxi a few times, taken a car service, and also taken the bus to & from LHR to Oxford, the bus does take longer. Good luck!