London would be hard to camp in or near as for most parts it's 15 -20 miles from the center to the closet countryside. Hostel or couch surfing e.t.c will be the only way. London has a good bike culture so if your into that it's there to find. The forum in my sig can be useful for that if you can wade through all the usual forum bull poo.
Riding from Heathrow, it's either the Grand union canal (
map link, google for routes out of heathrow on a bike) to an easy slow route, Uxbridge road for a medium route or the A4 for a just get there route.
South of the UK is fairly easy to stealth camp in many areas due to lots of small woods. There are some part though without much cover, many part of the north have this too, but in my experience the woods are bigger but less often.
There are regular bridleways (bike, horse and foot), blue arrow and byways, red arrow (horse and trap, horse, bike, foot and sometimes offroad car access) all over the countryside to get you away from the road, and countless footpaths, yellow arrow (not allowed to cycle along but you can push your bike along) again to get yourself off the road and down towards a secluded area. Just look out for the signs as you ride along a road, if a track does not have a sign with either one of the coulored arrows or a sing saying footpath, bridleway .e.t.c. unless you have a map showing it as one you probably have no right of way down that track. Places like Dartmoor have free camping rights, google again.
Campsites in the UK have cost me from £3.50 up to £12 (end of a day can't be arsed to find anywhere else).
Scotland you have diffrent camping and access rights. Google.
Bike on train in the UK and France (non TGV service) are generally fine out peak hours in my experience. You will have to remove your panniers for storage to make your bike smaller. Occasionally I have had to wait in the uk for a space to be available in the bike areas as they were full, but you can book a space if needs be. I hear heading out to the Western Isles in Scotland and right up to the North towards Wick this can be a problem because of high number of cyclists so best to book.
My experience of France is that it's similar to England, some areas are easy to stealth camp others offer little cover but there are lots of campsites, summer prices I have paid vary from 5 Euro to 12 Euro per person. Never camped in Spain or Italy, only ever stayed in cottages e.t.c.
Never camped in Holland so can't help but Amsterdam has excellent cheap hostels and hotels, nice caf
és, nice coffee shops e.t.c for that side of things but if you go defiantly checkout the Sex museum (cheap about 5 Euro I think), quite good fun and it's not just all porn, van gogh gallery, expensive but really worth it, amazing and the Jewish museum, interesting to see the full history of the Jewish side of the war. I hate religion but I still found it very interesting because it's all to do with the political and prejudices rather than any over the top zealous religious preaching.
Never been to Switzerland, all I've herd is that it's beautiful the people can be pretty unfriendly and it's expensive.
I have been told by many people Barcelona has a very bad pickpocket problem so be aware. Otherwise normal tourist rules apply, don't open your wallet bulging with cash, have two wallets e.t.c.
Contrary to marf's comment I've never been to the US only to Canada and only to Alberta and personally found public land and rights of way very hard to come across outside towns, everywhere had no trespassing signs when looking for a place to go for a short walk. We could find big open places miles away but never any just round the corner places. So it looked like it was good for drive ages to go on a big multi night trek, but bad for a afternoon wonder. So I guess the idea public access is diffrent in each country, lotts of righs of way here but less open access land apart from in national parks.
P.S It should go without saying any wild camping should involve taking away any litter and leaving no trace. If you can carry it in you can carry it out.