My missus came to work here from Spain, so I can tell you about NHS etc. As a European, you qualify for the same as and english person. Just register with a local doctor and there it is. I think it's all to do with the EU movement of trade which makes it easy for you.
What everyone else is saying - if you want a cheaper place, stick south of the river. Transport isn't so good as north London, but it's made up for by the difference in rent. We're paying about £850pcm without bills for a one bed here in a nice part of north London, which would be much cheaper elsewhere. Location is everything.
I lived in Wimbledon a while ago and it was pretty lovely, easy to get into central plus quite green. Also, reasonably near the M25 to drive out of London too.
Finally,
Gumtree is great to find yourself a place. We've let almost exclusively via independent landlords, which will save you a fair few quid in agency fees. You can filter your search for property by size/sharing/area. Property in London moves EXTREMELY quickly. If a nice place is advertised at lunch, they'll have full bookings by 3pm and it'll have gone by 7pm.
Good luck if it all works out :thup: