So in the United States, if you're a resident of another state, and earn income in another state (say, as a freelance whatever), the income you made in that state is owed in tax to the state in which you earned that income in.
The beauty of the 21st century is that, if you work digitally, you don't have to claim that work in the place in which you were located when you were working. What counts is where the location of the machines are where you saved your work. So as long as your bank is in the UK, your job is in the UK, and your resources are in the UK, none of those entities are paying US taxes, and neither will you.
Of course, if you're found out, you'll be looked into...but if you're not working at a brick and mortar branch of a UK company in the United States, you'll have a lot of evidence supporting your position.
Otherwise, your rules are cut and dry - on an F1 visa, you'll have to work for your school or government entity only, and anything outside of that is considered a violation of your immigration terms.