I've never done a TA, plus I'm a US citizen, but as I recall, it was a pretty quick process. With that said, your trip being a TA, plus not being a US citizen, may add a couple of hurdles to the process.
The fact that you are from the UK doesn't make any difference about CBP. Everyone needs to go through CBP. Assuming Cape Liberty is your first US port, you will go through as you leave the ship. Assuming you have a later flight, shouldn't be an issue.
Since your question isn't related to Royal Caribbean, you should post on the ports of call boards.
https://boards.cruisecritic.com/forum/99-st-croix/
https://boards.cruisecritic.com/forum/329-st-kitts-amp-nevis/
https://boards.cruisecritic.com/forum/82-st-thomasst-john/
As an IT Security guy, all I can say is that using outdated phones that are running software no longer supported by the vendor can put you more at risk for security issues. Just be careful.
Personally, while I hope it's available on stateroom TVs for those that wish to see it, I hope it's not available in public spaces. Not all of us want to watch it/have it forced down our throats.