The UK Government have released detailed guidance on demonstrating your COVID vaccination status.
You can access your COVID-19 vaccination status through the free NHS App from 17 May. You can access the app through mobile devices such as a smartphone or by tablet. Proof of your COVID-19 vaccination status will be shown within the NHS App. If you are travelling abroad, we recommend you register with the app before booking international travel.
If you do not have access to a smartphone and know that the country you are travelling to requires COVID-19 vaccination status, you can call the NHS helpline on 119 (from 17 May) and ask for a letter to be posted to you. This must be at least 5 days after you’ve completed your course of the vaccine. We expect the letter to take up to 5 days to reach you.
Further guidance can be found here – https://www.gov.uk/guidance/demonstrating-your-covid-19-vaccination-status-when-travelling-abroad