I suspect that when traveling to the USA your booking info may go to some branch of the TSA that automatically scans for prohibitions on your entry. NO clue if that happens in the UK or not.
Yes, the UK does.
The UK is given the infomation of everybody who books to land in the UK, either advance airline bookings or at the ports, eurostar. There can be people there to meet them if they are wanted, haven't paid their NHS debt, or embassy staff at the foreign ports to stop them boarding/remove their British passport*.
The UK also collects the names of everyone who exits the UK, either the advance bookings list from the airlines or from the ports or eurostar. This can either trigger details of those the UK will prevent from returning; or they have staff there to stop that person from exiting; or they use the data to see which visas cause the most overstayers and which country they are from.
With the UK's ETA (Electronic Travel Authorisations)* everyone will need an ETA to be able to board/enter the UK., unless they have a UK visa, or a British Citizen passport or Irish passport, unless there is an exemption.
* Were made law under the Nationality and Borders Act 2002.