How to let NHS know they’ve got the wrong address for their patient?

Take it physically to the local GP Practice as good chance it was one of their patients and may still be. GP Practices are where the "official" NHS spine addresses come from. Acute Trusts/Mental Health Trusts can't usually update this information and rely on the GPs doing so. Hospitals (NHS Trusts) often have a "contact" address in their EPR (Electronic Patient Record) that does not synchronise back to the spine, normally because the patients tell them to use this address , so the addresses get out of date/sync. Getting the address updated is the patients responsibility, and it HAS to be done by the GP, so they need to tell the GP to update it. If the Hospital tells the GP the address is wrong, the GP has to validate this with the patient and sometimes don't or are unable to , so it stays unchanged. If the letters have NHSBSA on the back this is something different and normally about dental charges, GHIC Cards or prescription pre-payment certificates (or NHS Pensions). Return the letter to them.