I'm terribly sorry to hear this, it doesn't sound fun at all

What I get from reading your post is that you have a dream to do Medicine, and it has gone a bit pear-shaped because you couldn't study Biology and Chemistry at A-level at your sixth form. You also feel like people are ashamed of you and are feeling jealous that they'll go on to uni while you take a gap year or go to a college and start afresh
First, let me assure you that no one is ashamed of you, and if you asked for help, you might even receive some at school! And taking a gap year while your friends go to uni is not so hard, from my own personal experience. It's a funky feeling at first, but it's just a part of life, and you'll meet other people in your gap year and then go off to uni and make new friends and everything will be OK.
Secondly, resitting your GCSEs might be a bit of a moot point. Unless you want to apply to a uni that scores GCSEs (a lot don't, other than asking for minimum grades), you can probably get away with just sitting the A-Levels for Biology and Chemistry in a gap year. In fact, if you pick your uni options right, you might even get away with sitting just one of them, although admittedly it's probably best to sit both of them to have as many uni options open. It's true that some medical schools get funny with A-levels if you don't do them all in a single sitting after two years of study, but not all universities will be like that, and most importantly, it doesn't matter! Your priority is to get into Medical School, the uni you end up at is irrelevant for Medicine - you'll get the same degree from them all, Russell Group or not.
It won't be easy, but if you really want to do it, you can still become a doctor. Don't give up. Carry on trying to arrange work experience and doing extracurricular stuff, and in the background research the universities you'll be able to apply to (it won't be all, but you'll have lots of options. If you put your mind to it, you'll get this. Good luck
