As a European who's currently living in North America, I feel that there's a lot of confusion over salary comparisons between different markets. Let’s break it down step by step:
First of all, think in one currency, even if the two currencies are close in value. 1 GBP is currently 1.2 US dollars. Or conversely 1 US dollar is 0.83 GBP. Convert all the numbers you see into one currency to get an objective measure of your expected personal income. Forgetting about currency conversion is especially prevalent for countries whose currencies are "dollars" - a Canadian dollar is worth merely 0.73 US dollars, regardless of what it’s called, but Canadians often make the mistake of comparing the two as if they have the same value.
Remember that the US is a huge country, so there are more cities to choose from than just New York or San Francisco. Just like there are cities other than London in the UK. It becomes especially important when you add taxation into the mix, as different US states have different tax levels. Working for a company in North Dakota won't yield you the same salaries as working for a company in California, so don't base your calculations about the entire country based on just one location.
If you're employed as an office employee at a major company, you have access to most of the benefits provided "free of charge" by the government in UK. Healthcare is always covered and is usually more accessible in the US than the "free" healthcare in the UK. For example, a person with a good health plan can usually get an appointment with any specialist within a couple of weeks, compared to waiting for several months at British clinics. It gets more difficult if you contract a long-term disease such as cancer, but as a European this is of very little concern as you can always pack up and move back to your home country.
Vacation days are a little worse than what you get in Europe. The average highly-paid office employee would have around 21 vacation days per year, compared to an average of 28 days in Europe. There are also a bit more public holidays in the UK. So you should deduct one week's worth of wages from your US salary to get the vacation-adjusted version.
Retirement benefits are theoretically better in the UK, but in practice you're extremely likely to get screwed by the time you retire as birth rates remain too low to support Europe's current pension system. In comparison, the US offers you numerous retirement saving options that are more reliable than your home country's pension fund, as long as you save up a healthy amount each year.
Moving with a family
If you have children, there are a couple of additional variables to add into the mix. In the UK, you can get free kindergarten starting from the age of 3. In the US free childcare only starts from the age of 5 in the form of pre-school. Elementary and high school are free in both countries and comparable in quality. University education in the US is more expensive than in the UK, though by how much will vary depending on the college and the scholarships you're eligible for. If you relocate to the US, your child won't be eligible for home fee status back in England unless they spend at least 3 years in the UK first - though even with "overseas status" tuition might still be cheaper.
Salary after tax
It’s very important to completely forget about gross salaries and focus on net (after tax) salaries. This makes a big difference as the total tax rate can range from 10% to 50%, depending on the country.
Let's compare net salaries at the same company in two cities across the ocean: London and Seattle. Seattle is not as big of a city as London but still has expensive real estate costs as seen below.
Google Seattle pays out an average of $148k in base salary and $68k in stocks per year, for a total compensation of $216k per year. This comes out to $158k per year after tax.
Google London pays out a base salary of GBP 74k and 13k in stock bonuses, for a total compensation of GBP 87k, for a net pay of GBP 56k or $68k per year after tax.
Now let’s deduct your real estate costs, as they form the biggest difference in living costs. Prices on food, electronics, dining, etc., are usually a small part of your budget and are comparable across all Western countries. You can also ignore differences in transportation costs if you compare real estate prices in the downtown area, as this way you can just walk to work in both countries.
A 3 bedroom apartment in downtown Seattle costs around $3.9k per month or $47k per year, leaving you $111k per year after real estate costs.
A 3 bedroom place in downtown London will also set you back $3.9k per month or $47k per year, leaving you $21k per year. If you instead rent outside the city center, you'll have $37k/year left over.
This is just one example of a calculation, as not everyone will be moving from the UK to the US to work as a software engineer for Google. Compare the total compensation in the UK with total compensation in the US for the same job to figure out which location is more profitable in your specific case. You can also use Numbeo's figures for a 1-bedroom apartment instead of a 3-bedroom one - or just lookup a local real estate website. In general, the US should always win for STEM occupations, especially if you're working for a gigantic multinational corporation like Google.
Once again note how important it is to compare two very specific locations rather than "UK" against "USA". Moving the comparison from Seattle to NYC will make you lose 12% on state income tax as well as pay higher prices for real estate. Sometimes it might make sense to choose a different US city to move to or just relocate from London to a cheaper UK city instead.
Use the above methodology to obtain the precise amount of money you will earn living in each location, taking into account your personal situation and living expenses. Using "rules of thumb" is unnecessary and can lead to incorrect assumptions. And as we all know, assumption is the mother of all screw ups. Spending a few days on building up a spreadsheet with all the nitty gritty details is a worthy time investment when it comes to making huge personal decisions.