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:
Currency
As a starting point make sure to think in one currency, even if the two currencies are close in value. 1 GBP is currently 1.28 US dollars. Or conversely 1 US dollar is 0.78 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.
Mental shortcuts are useful when calculating the tip at a restaurant but have no place for major life decisions where millions of USD are potentially on the line.
Location
Remember that the US is a huge country so make sure to not base your analysis by only looking at New York City or San Francisco. Seattle, Austin, Miami, Denver, Arlington, Boston and many other places have jobs available as well. Similarly there are cities other than London in the UK with a much lower cost of living.
It becomes especially important when you add taxation into the mix, as different US states have different tax levels. Dive deep into all the options, compare real estate costs, income taxes, real estate taxes, weather patterns, etc. Investigate the top 50 opportunities in America, not just the first one or two that you can find.
Benefits
If you're an office worker at a major US company you have low-cost access to almost all benefits provided "free of charge" by the government in the UK.
American healthcare is always covered by your employer (with a $100-$200/month copay) and is usually more accessible in the US than the NHS 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. You get to have your cake and eat it too: high upside from American salaries and yet you're still protected from the worst downsides by having the option to move back.
A curious reader might ask: what if I want to move to America and stay there for the rest of my life, rather than stressing a move back to the UK? Worry not, you still have excellent options to protect your downside risk. If you lose your job for whatever reason you will switch to a COBRA plan for up to 18 months to stay covered, which would cost you around $600/month. COBRA lets you keep the same coverage as you had while employed. After 18 months of being sick and unemployed you'll become eligible for Medicaid, so the American government will start covering your healthcare costs. Combine this with pre-purchasing disability insurance and you should be fine even in the worst health scenario. Things used to be a lot worse before Obamacare but those days are long gone. A modern day Walter White would not need to sell drugs to pay for his cancer treatment, as his insurance would've covered it all.
Vacation days are a little worse than what you get in Europe. The average American office worker would have around 21 vacation days per year (yes, legally speaking employers are not required to offer vacation days but in practice 3 weeks is standard), compared to an average of 28 days in Europe. There are also a bit more public holidays in the UK. If you care a lot about vacation days, reduce your American salary estimate by 51/52 = 2% to get the vacation-days adjusted number.
The UK has a government-managed pension system but anyone currently under the age of 50 is extremely likely to get screwed by the time they retire as birth rates remain too low to support Europe's future retirees. The US has a full equivalent called Social Security plus numerous retirement saving options like 401k and Roth IRA that should be much more reliable in the future as the government is much less likely to take away money saved in privately managed retirement accounts.
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 exact 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. If college is a big concern for you, I suggest making a list of 15 American colleges you think your child might be interested in, calculating the median tuition for each, and seeing how much money you'd lose by moving.
Salary after tax
It’s very important to completely ignore gross salaries and focus on net (after tax) salaries. And remember that different US states have different income tax levels, so this number will also depend on where you'll end up living.
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. [Yes, these are real salaries. You can also check Levels.fyi as another reliable source of salary information.]
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/bike 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 London to Seattle 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, doubly so if you get a job at one of the ~15 biggest tech companies.
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 perhaps relocate from London to a cheaper UK city instead.
Conclusion
Use the above methodology to calculate the precise amount of money you will earn living in each location, taking into account your personal situation and living expenses. Do not use "rules of thumb" under any circumstances. Spending two weeks on building up a giant spreadsheet with every tiny detail is a worthy time investment when it comes to making one of (if not the) biggest decision of your entire life.