Normally the GCSE and A level certificates (the official thing and not just a statement of results on the day) are issued around early November (after review of marking and final grades have been confirmed) and are sent to the school, who will email you or your DC to come and pick them up, bringing ID if the DC are no longer students of the school or if it's a parent picking them up.
All schools will say, if you don't come and pick them up by x date (they usually give you a week or two) they will be put in an envelope and sent to the last address DC gave them (they usually advise not to choose this option if you can, as certificates can get lost and they're not paying for registered post). They all say they don't keep copies (and they won't keep any that the pupils or parents forgot to collect). Besides, the universities usually want to see the original.
The exam boards will issue replacements at a cost (as always) if the certificated have gone missing if you apply for it. No idea how long each board takes. If your DC's GCSEs with more than one exam board, each exam board issues one certificate for all the subjects that were done with them for your DC.
The certificates generally look disappointingly utilitarian (even ugly, in fact) with a "quality stamp" from government organisations on each one, making them look more like electricity bills than an exam certificate. I remember relatives framing up beautiful O level certificates of their kids many years ago that looked very grand, with embossed pictures of wax seals, neat long lists of the subjects and grades, and an elegant signature, with the student's name smartly typed out. The current GCSEs cost more but the certificates look hideous in comparison, like an oversized utlity bill on stiff paper. That's what you need to look for, OP! Haha.
Might they still be in the A4 envelope most schools issue them in?