Hi OP.
Whatever you do don't get it done in Hungary. If you need a full clearance and completes that's nothing that a uk dentist won't do well (and on the nhs if you have an NHS dentist), if a uk dentist won't do it it's not something that should be done. There's no recourse for dentists in Hungary, no way of adjusting the work they do etc. Dentistry in the UK costs what it does because that's what ethical dentistry costs, and nhs dentistry is actually incredibly cheap for the work you get.
The problems with full dentures is uppers work well, lowers can be a pain because they rely on muscular control. The longer you are without teeth the more bone you lose and the less likely you will have a good fitting set of dentures. The less bone you have the poorer the fit
It sounds like you are only missing 4 teeth? If so that's really nothing. Do you have a partial? If not I would start with a partial and see how you get on. See how you get on eating and keeping it in. Find out how it feels. With the amount of teeth you have, and the lowers showing no mobility theres no way you need full dentures. Try a partial denture first and adapt to having a big block of plastic in your mouth
Dentures aren't really teeth, they don't behave like teeth and can't be used in the same way. They are a good replacement if you are missing teeth but the real thing is 100x better.
If your teeth stain there is a reason, do you see a hygienist? Book yourself in with a hygienist, or just your dentist just to talk through hygiene and risk factors of gum disease, causes of staining etc. Theres a reason your teeth stain, do you use corsodyl? Or smoke? And it can be sorted.
I really really would advise against a full clearance with only 4 teeth missing, and if you were my patient I'd be doing everything I could to help you get the staining and gum disease under control so you didn't have all your teeth taken out. What about a perio specialist? To get the gum disease under control?
I honestly think going from mostly natural teeth to a full set of dentures is an incredibly hard transition, and once your teeth are gone they are gone. Even if I thought that was the future for you, if you were my patient I wouldn't go it with full dentures, because it will be difficult to adapt.