My take on the cost of living is that it only seems high to an American who is still operating with, and from, the US dollar and thus having to keep the exchange rate in mind at all times.
When you mentally double every pound you have to spend, of course it's going to seem like the cost of living in the UK is outrageous. But it's really not once you are here and earning here and spending here, (aside from the universal woe of prices always rising, but that happens everywhere).
But what is overlooked, I think, is that, within the UK, to people living in the UK, and to people earning and spending and thinking in terms of only the Pound, it's actually exactly the same "cost of living" as the States, in terms of the value of items, the value of your earnings, and what you can buy with the unit of money you are talking about. Because, have you noticed a funny thing, that what it takes to earn, say, $500 dollars, and what that $500 can buy you in most of the United States, is actually the same for what it takes a person permanently living in the UK to earn or spend £500.
For example, taking a random item, I've noticed that a boombox I bought for the kitchen in the States, cost me about $50. Over here I was looking at the same kind of portable boomboxes, and they cost around.....£50! Now, to an American bringing their dollars over here, that's going to freak them out because in real terms that means $100 roughly. But think about it, to a Brit, living here, £50 is the same "feeling" to them as $50 dollars would be to an American in America buying the boombox I bought in America for...$50! I've noticed the same with other prices of things, and with earnings. It all actually has the same value, even in the numeric digits. I don't know if I'm the only one who thinks that is a strange numerical quirk, or maybe I'm just weird, lol. All I know is, it seems to me that the exchange rate, in effect, only makes a difference when one has to operate with it. (Which, I know, is still the situation for many here on this forum.)
This is all a longwinded way of saying, the cost of living in the UK is not bad when you stop converting to dollars, which will always cause a shock, and instead think of those pounds as having the same value when spent in situ by those in situ. Don't know if that makes sense as I know it sounds crazy, but...!