Jane wrote, "I'm afraid Premier League matches aren't the kind of thing where you can just roll up and get a ticket on the day." If you cannot buy advance tickets, this may change your whole schedule.
I would take the train from London to York. Without a doubt. Then pick up a car in York.
I know you said you'd like to get as far as Jedburgh after the game. But there are some great sights on your way driving north from Middlesbrough. Durham is an interesting city, with an amazing cathedral.
Slightly north of there is Newcastle-Upon-Tyne, the eastern start of Hadrian's Wall. Don't know if you are interested in driving west along the Hadrian's Wall for awhile, but it is interesting. And very historic. You could stay somewhere along the Wall at a cool B&B for one night.https://en.wikipedia.org/wiki/Hadrian%27s_Wall
WikiVoyage page for Hadrian's Wall; where to eat and sleep along the way:
https://en.wikivoyage.org/wiki/Hadrian%27s_Wall
I guess my point is, there are tons of things to see between Middlesbrough and Edinburgh. I wouldn't necessarily be in a big rush to get close to Edinburgh the first night after the game. Edinburgh sights can be done in one day, so you will be looking for other things to do and see, anyway.
However, if Jedburgh is your goal for the night, I would make a reservation now for something there. It is much easier to do your research now and choose something using an internet search than to drive around at night in a town you are unfamiliar with searching for adequate lodgings.
https://www.tripadvisor.com/Hotels-g319810-Jedburgh_Scottish_Borders_Scotland-Hotels.html
I assume all your airline tickets have already been purchased. If you cannot get tickets for the game ahead of time, and attending the game is knocked out of your schedule (hope this doesn't happen), consider taking the train from York to Edinburgh. York is a two night stop at the very least, if you have the time to spare in your schedule. Lots to see; Viking stuff, Roman and medieval walls,
medieval buildings. More interesting than Edinburgh, in my opinion.
You will not need a BritRail pass. You will need to buy your train tickets online in advance. Buy tickets London to York. If you conclude you are NOT going to the game, buy a second set of tickets from York to Edinburgh. If you buy now, online, you will save a bundle versus the price if you walk up on the day of travel and buy. Of course, you can buy that way also if you are unsure of what your travel days will be.
