The NHS App links through to your GP online access provider, in our case Patient Access, it doesn't store your data separately. So you could log into your GP records online and it would provide proof of your vaccinations there. But I suppose the advantage of the NHS App is that it just displays the vaccine status and nothing else on the front page.
However there are three problems I found. If you leave it open on the front vaccination page, after a while it switches to a Covid advice page on its own, and you can't get back easily to the first page, which would be hugely frustrating if you had just reached passport control and it disappeared!
Also the tabs to access to the rest of your records seem to be still active down the bottom, so that handing your phone over could result in one of these being tapped accidentally and your records being received.
The third issue is that it would be incredibly easy to mock up a vaccine status page and show that, I am sure people are working on that right now!