The weather will be iffy--but that can be true almost anytime, anyhow.
Whether your plans work for you depends on your style of travel and what you most want to see.
My first time in London, we had 2 1/2 days. In that time, we saw Oxford Circus, Piccadilly Circus, Trafalgar Square, Westminster Abbey (although, as it was a bank holiday weekend, it was horrendously crowded), the Tower of London (toured the tower and the crown jewels), the outside of Buckingham Palace, Hyde Park, including part of the Diana memorial walk, the Peter Pan statue, and the Victoria and Alberta memorial, the outside and some of the inside of Kensington Palace, the Abbey Road crossing, the Monument to the Great Fire of London (climbed it), the London Eye (rode on it), the Nelson Mandela bust, the Leake Street tunnel (cool graffiti), Carnaby Street (did some shopping), and a show (Billy Elliot). Mind you, we had spent the previous week in Paris and so we weren't jet lagged.
We spent 4 days (5 nights) in Edinburgh. On one of the days, we did a full-day tour of the Highlands with The Hairy Coo, which left us 3 full days in Edinburgh. We toured the castle, visited the Camera Obscura/World of Illusions (really cool, but we chose this partly because the weather was bad that day), toured The Real Mary King's Close, walked up and down the Royal Mile, went to the Writers' Museum, saw the fabulous Sir Walter Scott Monument (can't miss it really), did a Harry Potter Tour of Edinburgh, visited Calton Hill (twice--once in fog and once in sunshine), tried to visit Holyrood House, but it was closed that week due to visiting dignitaries, ate jacket potatoes in the library cafe downtown, toured the Royal Yacht Britannia, ate at the Sheep Heid Inn, said to be the oldest pub in Scotland, and generally walked around and looked at scenery and architecture. Again, we had been elsewhere in Europe (Denmark and Norway) prior to this, so we weren't too tired.
So, I say you can do and see a lot in 3 days in each city, provided you are not too tired or jet lag doesn't slow you down much, and you realize that you can't see absolutely everything. Also, be aware that Edinburgh can get pretty cold.