As regards car rentals, stick with the major names that you recognise; just be alert that some brokers may divert you to less well thought of options. If Green Motion or Easyrent were to appear, walk away!
If you are not venturing outside Oxford, may be consider if you actually need to hire a car.
Oxford is a small place, well served by public transport and taxis. And you may find it difficult or expensive to park your hire car in the city centre (if that is indeed your plan).
That said, I use Enterprise when I need to hire a car. Their staff will always give you a lift at the end of the hire period to the train/bus station.
Who ever you chose, look at their opening hours. You mention Sunday - I know Enterprise is open until 4.00pm on a Sunday, but I'm not sure all other rental companies are.
And again personally, I'd take the bus to Heathrow (as PMQ recommends).
I advise against using Hertz Oxford Osney Meade. Our pickup there was a disaster.
The car hire was scheduled to open at 10. We got there prior to opening because we had an appointment and were in a rush. The desk clerk was helpful and fast and I was pleased we were going to get out so quickly. HA. We got in the car and were ready to pull away when we realized the petrol tank was EMPTY. The lot staff came over and apologized and told us they'd get someone to drive our car to the petrol station and fill it up and bring it back. We sat waiting in the car for over a half hour. I had to go find out the matter. Apparently a staff member from the previous night's shift left with the credit cards they use to purchase petrol. They were trying to get in touch with him and were just leaving us sit. I told them we couldn't wait forever on the off chance they could get in touch with this person ad get them to turn up. They told me they'd mark the tank as empty and told us the petrol station was just down the road. Well it wasn't - and it our first time driving on the left - navigating the roundabouts with no knowledge of the area, unable to find the station and on the verge of running out of petrol. It was one of the worst 40 minutes of driving in my life. And of course we ended up dreadfully late for our appointment.
Also - we had a damage done to our car at the end of our trip when it was parked on the street. Hertz billed us an astronomical rate for easy to fix cosmetic damage. We are having to get our credit card company and a lawyer to negotiate it.