If you want to "do" Britain then investigate how you would get a
student railcard. Britain is not very big - it takes about 5 hours to get from London to Edinburgh or Glasgow by rail, and you can goggle out of the window all the time at the quaintness of it all.
I'd put you up for a couple of days if you made it to here (a Scottish island, Glasgow Central rail station to my front door would take you about 4 1/2 hours).
Just sayin' ...