Sorry to hear about your grandmother. I hope you can enjoy some more time with her before she dies.
On your other point, I am also in my early 30s. I had a private health check when I was 25, and get one every few years now as well. I get mine free through work so am in a slightly odd position. However I'll give you the following advice.
If you can be honest with yourself then at our age you (with a little help from NHS online) can probably do a fair self-assessment. A private assessment will mainly be asking about lifestyle - how much do you drink, smoke, exercise etc. What's your diet? Do you do drugs? It's that kind of stuff. If you're honest with yourself then you likely know your vices and paying a Dr to tell you them may jot be useful.
On top of that they'll weigh and measure you, and do some basic tests to see how your flexibility is holding up. Height and weight you can do at home and the Internet can advise on what is healthy. With flexibility - can you touch your toes? Can you do a beginners yoga video on YouTube? Can you touch your hands together behind your back. Oh the other good test is can you stand up from a cross-legged seating position without touching anything with your hands?
Anything above that, stuff like bone density scans, full heart rate measurements, prostate stuff, gastro checks - all that is extra and will only be done if you or the Dr have any particular concerns.