The chartered flight is currently scheduled to leave Muscat International Airport in Oman on Wednesday evening.
The government will end study visas from Afghanistan, Cameroon, Myanmar and Sudan, due to what says is abuse.
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.