One would have to win the H-1B lottery (~25% odds), find a job in February/March/April of a year; find a company that is willing to wait for 6-8 months, and move to the U.S. in October (if they win the lottery).
In tech today (esp. in the Bay Area of NYC), it won't be that hard to find a job and a company willing to apply for the visa and wait for 6-8 months, if you can show you're a good programmer -- but there's no way to avoid the lottery.
And there are European offices for many of the BigCo's. Microsoft, Facebook, Amazon, Spotify, and Google all have European engineering offices (just off the top of my head).
Finally, there's also remote work.
Really no reason to be envious of an actuary's £33,130 starting salary.
The way it works it seems is that companies are willingly to take the risk if you have proven yourself very well outside of america.
But I agree somewhat, that software engineering is not the field, where you are likely to be able to have a career.
Also: have noticed the sheer amount of sickening hatred of immigrants in this thread? These entitled pricks think that they're the only ones deserving of an opportunity to live and work in the US (because they were "born here"), and they think everyone else should be kept out. It continues to amaze me how much contempt and hatred there is for immigrants, here on HN.
For reference, starting compensation for a 22-year-old fresh CS graduate in the Bay Area is $120k+ (higher than department manager in your list). 5 years in as an individual contributor or tech lead you'd be doing $150-$200k. My roommate was in actuarial and got the hell out when he saw the difference. Now he's a 'data scientist' and doesn't need standardized exams or accreditation to get a job.