You need to check your math. Jul 19 - Aug 1 isn't 10 days. It's 14. 13 nights. And since the first and last days are transatlantic travel days, they don't count as time on the ground.
LONDON - EDINBURGH -BATH- LONDON- PARIS -AMSTERDAM - PARIS - LONDON
Have you invented a teleportation device? Or have you looked at any of the train sites to see how long it takes to get from one place to another? While physically possible to get to all these places, it will leave you almost no time to actually see or do much of anything once you get there. And will leave you exhausted in the process.
With 3 intra European transit days, even your first idea of London, Paris, Amsterdam is a very aggressive plan. And you would have to return to London by the evening of the day BEGORE you return home, losing another day that could have been better spent elsewhere. A multicity ticket into London and out of Amsterdam would have given you an extra day of touring, and saved the expense of a train back to London.
I agree with the others that you need to reconsider your plans. Bath is easily done as a day trip from London. Forget Edinburgh. The best you can do is 4 nights in London, 4 nights in Paris, 4 nights in A'dam, and the last night back in London. Don't forget that each of those stays will lose a half day due to travel to the next place, giving you only 3.5 days in each city.