First of all, my travels to the mountains often start by flying from Amsterdam to some other city, big or small, but if I can, I don't sleep there and continue traveling locally, by train, bus or any other means, to end up close to the mountains the same day. In doing so, over the years I've gathered a bit of experience on finding out how to get from a particular airport to a particular mountain area, but I haven't been to any of the cities you mention, not even in passing. To visit the UK, I don't like flying to London because it's too far from the mountains. I've either flown to Scotland or to Manchester - both have good direct connections from Amsterdam.
That said, Bristol is closest to London, and not too far away from the southern Welsh mountains. Nearby Cardiff is closer to the mountains, but also further from London. After Bristol I would look at the Liverpool-Manchester-Crewe area, by fast train just about as far from London as Cardiff is, and a couple of hours away from the northern Welsh mountains as well as the Lake District. Specifically I would look for a place close to a major train station. That way you have several options. And, using a fast train, even Scotland is possibile from there, especially if you can take Friday or Monday off as well.
If you need to be in London often because of your work, well, then you don't really have much choice. Even Bristol means a much longer commute than I would personally be willing to do regularly. Instead I would search for a nice place in London, if at all possible within walking or cycling distance to work as well as to a major train station, and travel to the mountains in the weekend. You can then go to Wales or to the Lake District by Friday evening, and return home late on Sunday. That way you have two full days in the mountains.
If it's not because of work, and the only reason to look for a base near London is because it's convenient to fly to Croatia, well, think about where you'll be going more often: Croatia or the hills and mountains in the UK. If it's the latter, then it would be less important how long it takes to get to London, wouldn't it? From the cities you mentioned, and considering proximity to the mountains, I would then put the greater Cardiff area high on my list.