Edinburgh to Oxford

1. Take the train, change at Birmingham, it'll take 6+ hrs - https://www.nationalrail.co.uk/journey-planner/
2. Yes, for an afternoon flight no need to go into London. The bus takes you straight to LHR so it's easier than the train https://www.oxfordbus.co.uk/airline

On Option 1, by train, it is a lot easier to change at York than at Birmingham.

As a date specific check (although it may still be awaiting notice of any engineering work)-

Edinburgh depart 1200 York arrive 1414, depart 1435 Oxford arrive 1745 or Edinburgh depart 1400 York 1614/1636 Oxford 1945

The train from York starts there, which is always a benefit.

(You could book the train from Edinburgh 30 minutes earlier if you wanted a longer change time at York).

You can only currently book the Edinburgh to York part, not the York to Oxford section which should open in about 4 weeks time.

curious why you will take the 3+ hour drive to Edinburgh when Glasgow Central (yes I know about the fire but the station will be back running way before the trip) is around half an hour closer and has Avanti West Coast trains which can whiz you down to Birmingham?

agreed on the second question - use the Airline coach

Thank you for the advice. As to why we are not taking the train out of Glasgow, the driver is taking them back to Edinburgh to catch a flight the next day, so we did not want to take them out of their way. I cannot tell from the map I have if Glasgow is on the way to Edinburgh; is it too far out of the way? Does York rail station have lockers to leave bags (it would be nice to walk around before heading to Oxford).
Also, did not know about the bus—we will take it straight to LHR Thursday morning.

It depends what route the driver takes to Edinburgh, but if he comes down Loch Lomond then he's passing on the central motorway within a mile or not much more of Glasgow Central Station, so it's barely a diversion- 15 minutes at the most.
The problem at the moment of saying come down from Glasgow on a Sunday is that there is a LOT of engineering work scheduled for the next 3 years between Glasgow and Preston with many line closures having to happen.
It'll be about another month before we know whether that is one of the closure weeks, with bus replacements in force.

No there is no luggage storage facility at York- you would need to use an app based service (such as stasher) to find one close by. Being a Sunday the Minster is only open from 1245 to 1430 (between morning services and Evensong)