If you’re old-school, like someone who still watches TV on a schedule, you can fill out the paper slip. Write your new address in block capitals (make it neat, like a love letter to a bureaucrat). Then, post it to the DVLA office at the address on the form. There’s no stamp cost the DVLA pays for postage (fancy, right?). Expect your new logbook in a few weeks. In the meantime, avoid driving like a maniac to avoid tempting fate.
How To Online Change Address In V5C Logbook Certificate UK - YouTube
What About the Car’s “Registered Keeper”?
This is different from the owner. If you lease the car or it’s on finance, the registered keeper is often the finance company. You must still update your address as the registered keeper if you’re the driver. If you’re not sure, check your finance paperwork. If you’re a secret agent, call your handler. Or just call the DVLA.