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I know people who do this full time in Bristol and find it ok, although the guy who I know who does it long term has a workshop space where he can touch base and park up out of the city and spread out.
Having lived and travelled in our van for six months this year I would hesitate to do it in London long term. In summer you basically can't be inside with the doors closed due to the heat, but if you're just on the street you can't easily spread out. Winter will get pretty claustrophobic too.
Why not get a boat? Boat living in London is great.
Edit: When we've been in cities on normal streets it does always feel a bit exposed, I'm sure you get used to it, but for me being in a van is about being in the middle of nowhere.
showers could be got around (gym,work), toilet would be more of an issue, no? wouldnt want to be caught short. plan would be to park it on quiet suburban streets.