The Spectator’s Fraser Nelson has written a good piece about the Swedish “free school” system – basically a system where private schools get money from the councils depending on their number of pupils.
AFAIK, Fraser is married to a Swede, so he’s normally fairly clued up on Sweden.
I’m not quite sure how the Swedish free school system differs from the Danish one. I do believe it’s existed in Denmark for much longer than in Sweden, though.
Although I might disagree with some of the finer details, it’s one of the (few) areas where I broadly agree with the Tories. It’s absolutely ridiculous that private schools are so expensive here. It just drives up house prices in the catchment areas of good public-sector schools, while making it harder for new ideas to spread.