It was the first such ruling in Britain.
The High Court of Justice, Queen's Bench Division, ordered Bank Audi and SGBL to make payments of about $1.1m and $2.9m to claimant Vatche Manoukian by March 4, Reuters reported.
"Bank Audi will abide by the ruling of the British court," an official from the lender said.
SGBL did not immediately respond to a request for comment.
Lebanon's financial system collapsed in 2019 after years of unsustainable financial policies, corruption and waste.
Banks imposed tight controls on accounts, including an informal ban on withdrawals of dollar-denominated deposits and limits on withdrawals in local currency.
These controls were never formalised with legislation and have been challenged in local and international courts, with mixed results.
A UK court in December ruled in favour of a Lebanese bank in a case brought by a depositor.
It ruled that the bank had discharged its debt to the plaintiff by issuing cheques for the value of his deposits.
Many Lebanese banks have resorted to paying dollar-denominated funds through bankers' cheques, which cannot be cashed out in dollars and are instead sold on the market at about a quarter of their value.
A week earlier, a French court ruled in favour of a saver living in France in a case she brought against a bank that issued cheques for her account balance.
The ruling said the move by the bank, which was opposed by the claimant, meant the lender had not fulfilled its obligations.