Bereaved people hit by the National Savings and Investments (NS&I) lost funds scandal should be contacted by the bank in the coming weeks about accessing their loved ones' money.
Thousands of people were not paid all the money that their deceased relatives had kept with the bank, due to errors in identifying all NS&I products held by the customer before they died.
The estates of some 34,000 deceased people were affected, with the bereavement claims totalling £367m.
The former boss of the government-backed bank resigned over the scandal in March.
NS&I said on Tuesday that all affected estates with holdings of £10 or more would be contacted "reunite them with the full value of those holdings".
The funds will start to be paid out in the coming months and should all be paid out in the first half of next year.
Affected holdings will be adjusted upwards to include either the interest accrued since the error occurred, or the Bank of England base interest rate plus one percentage point, whichever is higher.
The bank said the funds will be exempt from inheritance tax and income tax.
Those affected will be given details of the process for requesting reimbursement of reasonable legal costs caused by the delay in accessing the funds.
Affected families, beneficiaries, and executors do not need to do anything at this stage, the bank said.
NS&I has said that the error that led to the scandal has been rectified, and a more robust process for bereavement claims was brought in in January.
Jennifer Brough, 82, from Doncaster, says NS&I needs reform because its service is "not good enough".
It took her six months to make progress getting access to her husband's savings following his death in 2024.
She said NS&I wanted her to provide a grant of probate, a type of legal document, even though her solicitor said this was not required.
"They would not release the money until I produced this," Jennifer said.
"I found this very hard. At a time when you are low, having to get help to do it for me to save money, eventually being paid, with a lot of unnecessary work."
Rhona Edwards's husband Alex passed away in November. After she gained probate, she completed all the paperwork to release his £50,000 worth of premium bonds.
By March, she was still waiting to hear from NS&I.
Rhona, who lives near Chepstow in Monmouthshire, told the BBC that the delay in releasing the bonds was preventing her from finalising and closing Alex's estate.
NS&I has 24 million customers and offers a range of saving and investment products, including premium bonds.
The attraction of its products is their security, with the safety of savings guaranteed by the government.
But some bereaved families have spent years trying to access their late loved ones' money, telling the BBC of multiple forms, numerous phone calls, and great distress.
Some were reportedly forced to turn to lawyers to help recover the money, incurring additional costs.
Sir Jim Harra, the interim chief executive of NS&I, apologised to those affected, saying: "Beginning the process of repaying these funds is a key step in putting things right.
"We need to ensure that everybody who makes a bereavement claim with NS&I is treated sympathetically and has their case processed as quickly as possible.
"Today, this process is taking longer than it should. We have brought in additional staff to get the service back on track."
