For gawd's sake don't start in the UK in April - it'll probably snow on you!
Greece, Balkans & coastal Italy are all too hot to climb in mid-summer so start there.
Austria is pretty cool, join the Alpen Verein to get acces to huts and insurance.
The highway prices in France & Italy are crazy! In Austria you buy a weekly/monthly pass - same in Croatia I think, Czech Republic & Slovenia too from memory. And Germany is still free but may change for foreign vehicles soon.
Dolomites, Chamonix, Switzerland and Alps are good options for peak summer.
Be warned - dirtbagging is super difficult - there's no free camping like we have - it's all caravan parks and motels. You get the occasional quiet campground in the woods - but even then there'll be some Danish campervan parked next to your tent.
I've heard great stuff about Corsica but haven't been - sounds like a real adventure.
Getting a camper van may suit best for the length of time you'll be there. Most places you'll need to check in to a campground so buset accordingly.
I've done Euro-climbing via public transport & car - the former was probably more interesting - the latter definitely more convenient. I have family over there so didn't have to buy the car. Remember the Uk drive on the left - so if you buy there you'll be forever confused on the mainland.