The Oyster Card is for travel in and around London; it's not something you'll use to get to Edinburgh.
The under-25 rail card may help you, or not, depending on the mathematics. Rail fares in Great Britain are among the highest in Europe if you just waltz into a train station and say, "I want to go to Edinburgh today." If you buy your tickets way ahead of time and there are bargain-priced "Advance" tickets still available, you will pay much less--maybe so much less that the cost of your under-25 rail cards would be higher than the discount they provide. However, those "Advance"-fare tickets will be for a specific train (date and time), so you need to be sure the trip is going to come to fruition on the schedule you anticipate.
What you need to do is go to the website nationalrail.co.uk and check the fares to Edinburgh for today/tomorrow and again for as close to your travel date as you can get, which I think is going to be some date in June. It's best to choose the same day of the week as the day you hope to travel, if you already know it. I think you'll see a very wide range of fares extending almost to 200 GBP per person one way for travel today, whereas there will be Advance fares well under 100 GBP for travel in June.
I don't know what a week-long travel pass would cost, but its financial utility to you may depend on whether you'll be confident enough to buy a date- and time-specific ticket to Edinburgh when tickets for your travel date become available. Fares for tickets bought way in advance will nearly always beat the cost of a rail pass.
Do you anticipate taking just that one long trip, or will you also return from Edinburgh to London by train after the driving part of your journey?
