You are increadibly nieve to think we'd really get a maglev line for that price. Never gonna happen. Unless you want to close down one of the existing north south lines whilst this is built. Even in Shanghai it cost over a billion dollars, and labour is increadibly cheap relatively there. That line was only about 20 miles. Prices of land particularly and labour costs are way more here, and proposed maglev tracks are a substantial distance longer.
Commute times in London are by enlarge longer than any other part of the country. London is many times denser, and thus make public transport a lot more cost effective, and that is how the government work these things. A ten carriage
I personally don't think it should be a case of London getting this or Manchester getting that or leeds the other, but every town getting more public transport investment. I think there is a good case for city metros having more autonomy over public transport (though that of course means there should be less national tax if local taxes are going to go up). They should have the abiliity to borrow (maybe from the government or banks or bonds) to invest in public transport. Maybe there could be a matching scheme. Maybe EU a third, national governmetn a third and the rest ogt by any other means (local taxing borrowing or private investment).
I know other cities suffer from over crowding, but I'm not sure how many other towns have to close off platforms every day due to dangerous levels of over crowding like Victoria, and London bridge (and probably many other London stations) do almost every day. Or where you have to wait for several trains to go past before you can squidge into the train.