Can't offer advice for continental europe, but here's some UK advice.
Weather is generally a problem. I'm visiting friends and family in the UK in June, and hopefully I'll get to climb outside. If not, at least I'll catch up with people and visit some of the gyms I used to go to. If you want guaranteed good weather, best pick another country.
Peak District is your best bet for Grit. There's bunkhouses at Hathersage and Bamford (Bamford one used to be really good - hopefully still is) that are around £10 per person per night. You could walk from both of these to Stanage Edge, though could also hang around the Outside cafe in Hathersage and get a lift from other climbers. Check out ukclimbing for other crags around. Couple of climbing centres in Sheffield for when it's wet (The Foundry and The Climbing Works).
Both Hathersage and Bamford are accessible by train, but 'buy-on-the-day' train fares in the UK are expensive. Take a look at www.nationalrail.co.uk for trains, and www.nationalexpress.com for buses (takes more time but cheaper).
I also found renting through Hertz that the prices can vary depending on the pick up office - Heathrow airport was double the price of picking up from Finchley (north London). Not sure if it's the same with other rental companies.
North Wales - only visited once, but once was enough to know mountains are a rain magnet, but the climbing on Anglesey (Holyhead mountain, Gogarth) can be dry whilst it's pouring around Snowdon. Llanberis is probably the focus of North Wales climbing - couple of gear shops in the town, and DMM have their factory there. There's a good small climbing centre in Llanrug. I had a car, but got the impression public transport would be harder to use here (though could be wrong).