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I'm in Reading, but much of the same issues apply.
A boat would, I think, be my ideal situation, but much more expensive outlay.
moocher-
You could find liveable boats within your budget. I lived on the water for about two and a half years. Bought a wreck and did it all up ourselves but £21k should get you something liveable. Might need a little work but if you're thinking of living on a boat I assume you're not afraid of a little DIY.
I work in Reading - happy to talk sometime about it all if you want to ask questions etc.
There are some difficulties with being of no fixed abode, mainly post but anything you need an address for like bank accounts becomes harder.
Lots of similarities between boat and van living I think. Practical things like toilet, power, water etc need consideration but if you're in Reading you can take advantage of water points along the canal (keys cost about a fiver), and elsan points for your poo.
Arducius
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.