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Wouldn't they need to get some kind of court approval to use NSO software on an American?

I'd be pretty confident that the software doesn't ask you for a PDF of the warrant before it does its thing

No, they can just ask the UK to spy on that American. ...Or all of them, even.

The UK then conveniently has an agreement to share intelligence it has gathered with the US.

If the UK shares intelligence that happens to coincide with a person of interest even if they are American, well, shrug.