At one time you could request names and addresses of previous owners of your car. Since changes in privacy laws this has been restricted to just supplying the date of ownership change and the town of the new owner.
To request this information you need to complete the form V888 downloadable from the DVLA website and post with a £5 postal order to the DVLA. It took about a month to get the information but they could not track (or lost) details of the first few owners!
It was interesting how the car had moved around the country in it’s 40+ years history see https://mymgbinfo.wordpress.com/my-mgb-a-brief-history/ to read a summary.