-
Is any of that illegal?
sex with a 17 year old in London,
No
New York
Yes
and Barbados.
No
I can't pretend to understand international law that much, but even if he were proven to have nobbed a minor in the US (by their standards), he would not face charges in the UK.
The US could seek to extradite though - And the UK extradition treaty with the US is such that a UK citizen will (not even may) be extradited for an act that is not criminal in the UK, and without prima facie evidence.
He's accused of having sex with a 17 year old in London, New York and Barbados. Is any of that illegal?