I did the London & Edinburgh double last year. I happened to go on a training run 2 weeks after London run by Paul Evans (Britain's last male winner of a major marathon) and asked what I ought to do between the two. His answer was: "Wrap yourself in cotton wool!"
As he pointed out, the effort of London takes a lot out of you, and also counts as a last really long training run for Edinburgh. You will not pick up fitness gains from training in between but will probably pick up injuries. So, on his advice, I ran no hard sessions, and long runs in the 6 weeks in between the 2 maras were: London Marathon, 5, 8, 12, 8, 6, Edinburgh Marathon.
On this plan, I still felt tired as I set off at the start of Edinburgh, and held back cautiously for 16 or 17 miles. Then, once I reached the 18 mile point where you turn to head back to the finish with a slight downslope and a following prevailing wind, I felt the freshest I've ever felt during any marathon at any point. I turned my new PB from London of 3:59:55 into an even better PB of 3:58:24.