I feel you. I have had the same group of friends since sixth form, and I feel immensely lucky to have them because (as is often the case) we met by chance and if we never met...who knows what I'd be doing now
When you say make friends 'at this age' I assume you only mean 18ish? Believe me, it gets a lot harder to make friends than it is now and, if anything, the next 3 years (of university, I assume?) will be the easiest time to make friends before work, distance, and generally declining time for socializing set in. Even if not, whilst a gap year might be a quiet time, finding some social activities locally or getting a job could be a good start for you
As far as I'm concerned though you're absolutely right to not want to force anything, but that means the only real way is to put yourself out there (socials, sports, clubs, in class) and hope to strike connections
The element of your parents moving is a really tough one and unfortunately, you can't control it but hopefully for you that is about to change. As an adult, you have the deciding power when it comes to staying put once you can support yourself and that should be good for you. To have moved so many times and still made friends at each place is a feat, especially in the UK. I'm not sure how this matches your experience but I find Americans are much more outgoing and inviting to strangers (although I also find them flakier and less likely to form long term friendships) whereas Europeans are better at long term friendships but its more exclusive