British customers of the Laiki Bank are instead reliant on the Cyprus Deposit Protection Scheme, which covers up to 100,000 euros or around £87,000.
A spokeswoman for Laiki Bank UK said there is "no sign" of its 13,000 depositors panicking about their money and insisted it is "business as usual" for the bank.
"We are sitting on a sea of cash," she said. "Withdrawals have been very, very low."
Those with money held at its three branches in London and one in Birmingham are not covered by the UK's Financial Services Compensation Scheme (FSCS), which protects up to £85,000 of customers' money if a financial institution goes bust.
The spokeswoman for the bank, which has branches in Mayfair, Palmers Green and Finchley in London and in Erdington in Birmingham, said that anyone who has concerns should get in touch.
But she added: "It is business as usual. Customers have been dealing with the bank in the normal way. There is no panic."
Cyprus needs to have come up with a roadmap by Monday as the European Central Bank has said it will cut off emergency support to the banks, which could trigger their collapse and end up with the country being forced to leave the euro.
Under European law, all European Economic Area countries must provide 100,000 euros in protection for the total deposits of each eligible saver.
The Cyprus Deposit Protection Scheme has been in operation for 13 years and it would come into play if a determination was made by the Central Bank of Cyprus that the bank is unable to repay deposits.
In theory, payments would then be made within 20 working days, although the Central Bank of Cyprus can extend the repayment period by up to 10 working days.
Customers of Bank of Cyprus UK, another Cypriot bank which operates in Britain, are covered by the UK's compensation scheme because of the way it is structured.
Prime Minister David Cameron discussed the financial crisis in Cyprus with Russian president Vladimir Putin in a phone call this morning.
Cypriot efforts to clinch a contribution from Moscow appear to have failed after Russia's finance minister was quoted as saying talks had broken down. The Kremlin has a significant interest in the situation, because Russian depositors are believed to have around £17 billion in Cypriot banks.
Downing Street said Mr Cameron and Mr Putin "agreed that they hoped a solution could be found to the crisis", but added that Cyprus took up only a small part of the 30-minute call, which also touched on Syria, the upcoming G8 and G20 summits in Northern Ireland and St Petersburg and BP's Arctic oil deal with Russian company Rosneft.
A new package is necessary after the parliament in Nicosia rejected a plan this week to grab up to 10% of bank deposits. Banks have remained closed for fear of queues of depositors withdrawing their savings.
The UK Government has said it will guarantee the savings of British Armed Forces personnel and civil servants posted to the island, and on Tuesday an RAF plane carried one million euros (£850,000) to Cyprus as a "contingency measure" for troops and their families in case cash machines stop working.
Asked today whether any help was being considered for other British nationals, a Downing Street spokesman said: "Obviously we understand that developments in Cyprus will be worrying for British nationals with savings in Cyprus. We would encourage British nationals who are experiencing difficulties to contact their bank."
Anyone planning a holiday in Cyprus over Easter should keep in touch with Foreign Office advice, which is being kept under review, said the spokesman.
After updated advice yesterday, the Foreign Office website does not caution against travel to Cyprus, but warns: "Cyprus is implementing measures to protect its banking sector. Check with your bank for further information. "
The government of Cyprus has announced an extended bank closure. ATMs, debit and credit cards can be used as normal; however, while banks are closed, we advise taking sufficient euros to cover the duration of your stay, alongside appropriate security precautions against theft."
The Downing Street spokesman said the UK regards the situation in Cyprus as "essentially a matter for the eurozone".
He added: "The situation in Cyprus remains fluid, but it is in all of our interests that Cyprus tackles its difficulties and its banks are stable.
"We will keep the situation under review."