The Bank of England's new Prudential Regulation Authority (PRA) has announced that all those whose accounts are in credit will be automatically moved to Bank of Cyprus in the UK.
This means that up to £85,000 of their deposits will be protected under the UK compensation scheme.
Laiki customers in Cyprus face losing much of their money above that amount.
It follows the country's bail-out deal with the European Union, under which customers with more than 100,000 euros in their accounts face a levy of up to 60% on the remainder of their deposits.
But 15,000 Laiki customers with an estimated £270m in their accounts in the UK are being told their money is safe.
They will now be able to access it via Bank of Cyprus UK. Unlike the UK arm of Laiki Bank, Bank of Cyprus UK is a full UK subsidiary and is regulated by the PRA.
On average, Laiki customers in the UK have £18,000 in their accounts.
About 5% of customers have more than £85,000, according to Bank of Cyprus UK.
Any money above that amount would not be guaranteed by the UK compensation scheme, but the Bank of England confirmed that all deposits had been moved to Bank of Cyprus UK.
"This is a good thing for customers," said a Bank of England spokesperson.
"Some of the deposits may be uninsured, but they have all been transferred," she said.
Speaking at the launch of the PRA, the Chancellor, George Osborne, paid tribute to one of its first decisions.
"In your very first day in existence, you sorted out a banking problem without having to come to me and ask for British taxpayers' money. And long may that continue," he said.
Authorities in Cyprus are in the process of winding up the Laiki Bank. "Good" parts of the bank are being transferred to Bank of Cyprus.
UK customers of the Laiki Bank whose accounts are overdrawn will not be allowed to transfer their money to Bank of Cyprus UK.
Their accounts will be frozen, said the PRA.
Those with mortgages or loans from Laiki Bank will see them transferred to Bank of Cyprus, not in the UK, but in Cyprus itself.
Such customers are advised to continue making payments as usual.
Those whose current or deposit accounts are being transferred to Bank of Cyprus UK are being told to contact their new bank with any questions.
Thousands of customers with funds in the UK arm of Cypriot bank Laiki will escape any levy on their accounts.