Best way to get from Oxford to Edinburgh on a Sunday

Train, of course. Just check schedules and see what works best for you that day.

what date are you travelling?

there's a lot of engineering work taking place in the Oxford area

http://www.nationalrail.co.uk/

Obviously take a train from Oxford. The question is which route, via London or via Birmingham.
Not via London, because that requires crossing London on the tube from Paddington station to Kings Cross station.
Via Birmingham may be slower, but it will be easier.
And, as always if travelling on a Sunday, beware timetable changes due to maintenance work.

You will of course find it cheaper to pre-purchase your train tickets as they rise in price as date approaches - though even buying the day before can save money. (Short journeys - you can pay on the day). You may find it even cheaper if you split the ticket at Birmingham. www.nationalrail.co.uk

We are traveling on July 8,2018- a Sunday.