Will you have a hire car? and do you have accommodation booked in Edinburgh?
Will you go in a car, or by train?
Car is not a very pleasant journey as it will be motorway all the way, and during August the roads will be busy with people on holiday.
Train is possible as you can do London to Lake District (though the trains from London don't do directly into the Lake District) and then a train from Lake District to Edinburgh.
The only station inside the Lake District is Windermere, but to get there you have to get off the train at Oxenholme then get the train to Windermere. The other option is to get off the train at Penrith and then get a bus or taxi into the Lake District.
Note the Lake District gets VERY busy in August so you need to make sure you have booked somewhere to stay before you go.
Edinburgh is EXTREMELY busy in August - you need to sort out accommodation YESTERDAY
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