My mum passed away in June. She had a £5 bond and I was given the option of cashing it in then or keeping it in the draw until June 2021 when it will be re-paid. As far I know there was no option for transferring ownership of the bond. Would be lovely if it comes up before next June!

I found this on NS&I’s website on a page about support during a bereavement which might be useful: “If the NS&I savings you are claiming include Premium Bonds, you can continue the holding in the deceased’s name. Bonds will
remain in each prize draw for up to 12 months after the date of the customer’s death. To keep the Bonds invested, you’ll need to send the Bonds to us
along with the completed claim form. Bonds can’t be transferred
into another person’s name but will be kept on record for 12
months and prizes will be sent to the person who is entitled to
the money. If the Bonds can’t be found at the time of the claim
but are found at a later date, you’ll need to destroy them. At the end of 12 months, the value of the Bonds will be paid out to the person who is entitled
to the money. You can apply for repayment at any time during this 12-month period. If you choose to have the value of the Bonds repaid immediately,
the Bonds will be eligible for prize draws only for the calendar month in which the repayment takes place.”