Yeah, that's a tough one to nail down. Two weeks ago Germany and Belgium were sweltering (nearing 100 F), and last week, when we drove to Belgium specifically to go to the beach, it was raining and in the low 70s. This week, now that we're home and safely away from the seaside, it's sunny and pleasantly warm.
Not that I'm bitter...
But to better answer your question: this year in Germany, with the exception of one week in April, it was cold, grey and rainy from November through May. Seriously. It was awful. We lived in Lithuania the two years before that - it's a good bit north of Czech, so hopefully you wouldn't have as cold weather as we had there, but I would guess that throughout Germany and Czech in March-May, you'd be looking at cool, wet, fairly unpredictable weather. I spent 10 days in April in Poland some years ago, and dealt with everything from (relatively) warm sunny days to snow. Again, a little farther north than you're planning to be, but typical of the continental weather pattern.
Last time I was in the UK was March 2004, the day we arrived was gloriously sunny though chilly, the next day it snowed, next day rained, next day warm and sunny...you get the idea? Even Italy, Spain and Greece are unpredictable in the early spring - we've spent the last three Easters in each of those countries and the weather has ranged from sunny and warm (say 70 F) to rather wet and chilly - and we had the full range of weather on each trip. Italy last year was the most reliable, but it was also the latest Easter, being toward the end of April. The other two were in March, and though we were able to paddle in the sea in southern Spain on a couple of days, it wasn't really pleasant for swimming/basking.
If you want to have relatively similar temperatures throughout, start in the south and work your way north. If you definitely want to experience the beach culture, plan on visiting the southern countries later in April or in May to get warm but not scorching days. The trade off is that you'll be colder up north, but the beaches will seem even more appealing then!
There are a lot of benefits of traveling in the off-season, so cooler weather shouldn't necessarily put you off visiting - after all, northern Europe isn't usually a top beach destination anyway - I'd be inclined to do more sightseeing in the north to begin with, and work my way south, planning on a more relaxed, beachy vacation at the end of the trip. And you'll still end up missing the high-season crowds at the beaches, so best of all worlds, really.
Do definitely bring along waterproof shoes, warm socks, a fleecy layer and a water/windproof jacket of some kind, as well as a warm hat. They don't take up a lot of room (you'll be wearing the shoes most of the time) and make a HUGE difference to your quality of life on a cold, wet day.