Alternatively, if you have the NHS app installed on your 'phone (NOT the NHS Covid 19 app), once you have signed in, it shows your vaccination record and you don't have a piece of paper to misplace. Obviously, if you have an aversion to Government apps, you will have to stick to the paper one.
Here is the advise from which you will see that the app is by far the least onerous of the alternatives. The little appointment card is NOT valid! https://www.gov.uk/guidance/demonstrating-your-covid-19-vaccination-status-when-travelling-abroad
Terry.