For the UK vaccination program, you can access the NHS COVID Pass through the NHS App. This is not the same as the NHS COVID-19 app, which 'pings' if you come into contact with someone who has tested positive for coronavirus. The NHS App can be found here.
To use the NHS App, you will need to create an NHS log-in, which will link the app to your vaccine record. For the international travel COVID Pass, you will need to verify your identity. This process involves taking a picture of your photographic ID and recording a video of your face. The lack of transparency surrounding this process has raised some questions about how this information is used. There is currently no evidence that this data is used unlawfully. However, if you do not have a photographic ID or feel uncomfortable using that system, there is an option to verify your identity without using a photographic ID or recording your face.
Note that this is separate from the domestic COVID Pass, which can be used for venues within the UK. For a domestic COVID Pass you simply need to provide your mobile phone number, which will be matched to the mobile phone number on your GP records. You cannot use a domestic COVID Pass to show your vaccine status for international travel. However, you may need to use this pass to show a medical exemption (see below).
You can also access a version of either Pass through the NHS website. If you prefer, you can also request a paper copy. You can find more information on that process here.
The COVID Pass does not give the person scanning it access to your medical records. It shows whether you are considered fully vaccinated or not.
If you were vaccinated outside of the UK, vaccination programs in other countries each have their systems to prove vaccination. More information can be found here.