My wife was saying the same, and mine is not on the NHS app. Googling it appears that the government is not going to (to date) include third (booster) jabs on the NHS app. So I suppose the only way of proving is to keep the card given after the booster jab.
Edit. Thinking about how incompetent...
Did mine yesterday and before the 182 day guidance, and contrary to earlier advice on here Churchill Sq have been doing walk ins for weeks. I had an hour wait around midday, friends went early morning and were in and out.