There's been a few articles in newspapers in recent years along the lines of insurance companies don't make as much as we think, personal injury claims are hacking great lumps off their margin, perhaps even to the point of loss. Okay, I admit being a cynical bastard I took that with a large grain of salt - I doubt the directors are poverty stricken, and they have the easiest solution in the world. Put the premiums up!
Still, if there's some truth in that then maybe the insurance compaines would actually welcome the 3rd party bit being taken off them. I'm only guessing here, but if you could divvy a premium up into £x being the third party bit, £y being fire and theft and £z being everything else covvered by a fully comp policy, I'd have thought the profitable areas are likely to be y and z. Like I say, that's a guess on my part and I'm prepared to be shot down in flames by someone who knows better.

But if my guess is right then the insurance compaines might welcome the idea of being able to concentrate on the most profitable areas, and let the government supply the basic 3rd party insurance as a public service.
As a public service the government should aim to run it as more or less non-profit, but would obviously retain any money that doesn't go out on claims. As I said before, that's their motivation for genuine safety improvements in the road network, though I would hope that they'd reinvest any surplus in the roads or the policing of the roads, rather than chuck it at the next hare-brained project that gets dreamed up over expensive brandy in the House of Commons bar. Not sure how the nuts and bolts might work. For instance, what do we do about rewarding safe drivers who don't cost the fund money? At the moment we have NCD, so could this go over to an insurance "tax" disc? At the moment VED is cheaper on cars that the government has decided are green enough (bag of bollocks if you ask me, but that's another argument altogether), so it wouldn't be unprecedented. But if driven by someone else or sold, what then? I think it would probably have to be either a flat fee for a given type of vehicle, or the disc would have to relate to the driver instead of the car. That might not be such a bad idea, but then would we have to faff about swapping discs when changing drivers? Flat fee for all vehicles wouldn't be fair eactly, but it would be simple.