Train from Oxford to London

You can buy on the day as trains can not sell out. But, depending on how far you buy ahead (for a specific timed train) you will pay half or less of the walk up fare.
On other days of the week you may sometimes pay less than 1/4 of the walk up fare, but Saturday is the busy day for days out, so Advance Fares are a bit higher.

mmaryallen, are you going into the city of London itself, or to the airport?

Trains from Oxford to London go to either Paddington (the fastest, main route) or Marylebone (the slightly slower, newer Chiltern Railways route). They're very close to each other, so unless the extra 20 min of the Marylebone route or the 5 min walk to the Baker Street tube station is a dealbreaker, check both for the cheapest option.

There is a bus line, the Oxford Tube, between the two cities with frequent service: https://www.oxfordtube.com/. Don't let the name Tube confuse you as it is a bus. Depending on where you are going, you might not have to go to a main terminal in London to get on the Tube. In 2017, I got on the Tube on a relatively small London side street. The trip took about an hour one way. If you want, you can book your ticket in advance or pay as you board.

I took the Tube to see CS Lewis's house on the outskirts of Oxford. There is a Tube stop relatively close to the house. And there is a Tube stop in downtown Oxford.

Here is my account of stopping at CS Lewis's house: https://community.ricksteves.com/travel-forum/trip-reports/i-too-was-surprised-by-joy

Going to the city of London to join RS tour

We did this same route this past November. Dropped our rental car in Oxford, took a walking tour of the city and University (kind of one in the same), then a late afternoon train from Oxford to Paddington. Easy peezy. Frequent train service and we bought our tickets at the station. In our case, we took a taxi (lots available) to our hotel, but depending upon where the RS tour begins, maybe transferring to the tube will get you close enough to walk to your meeting place hotel.

If you tell people where your hotel in London is, you will probably have people give you advice on best route. RS normally has his hotels close to a tube station.