Sir Keir Starmer had refused to grant the US permission to use the Diego Garcia military base.
The prime minister said the "UK is fully committed to the security of Cyprus and British military personnel based" on RAF Akrotiri.
Air travel in the Middle East has been severely disrupted since Saturday, with thousands of flights cancelled.
There are measures, announced ahead of the chancellor's Spring Statement, yet to take effect.
A court hears that an art consultant working for the TV hosts took "secret and unauthorised profits".