Charon,
firstly the weather in all of those cities is roughly the same. If it's a rotten British summer, it's a rotten French and Dutch summer too,(the reverse is of course also tue).
That said, France is a "good" weather city, it's beautiful, and the boulevards are made for strolling.
London is a "bad" weather city. The Brits haven't quite caught on to the European bistro terrace yet, and we're used to entertaining oursleves in poor weather.
The weather is irrelevant in Scotland, the landscape will impress, the towns depress (with the possible exception of Edinborough) irregardless of climate.
Amsterdam is also weatherproof, inasmuch as if it shines it's beautiful, and if it rains there's more than enough musea, galleries and restaurants to divert you.
Your chances of good weather in May or September are actually about the same (we have rotten springs, sometimes), but I'd recommend September. It's still considered the "season", and everything should be available, whereas May is often considered to be Spring, and some Summer activities or facilities may not be available.
Finaly, all of Europe is 2 hours wide by jet. You'll enjoy Northern Europe what ever the weather. But if you don't, Southern Europe is ónly an averagly good magazine read away...