This message is also to OP, @cantcopeanymore0 and anyone else going through these problems:
You say your GPs won't prescribe anti-depressants/mood stabilisers/the like for patients under 18. I completely agree with @Labibibabibidum that that is ridiculous. Teen brains are not exempt from needing psychiatric meds!
I lived away from the UK for twenty years, and at a couple of points when visiting, I needed a medication refill. Couldn't use my old GP as I'd been away too long, and this situation was what led me to discover the joys of the private GP. It's "only" about 120 pounds for an appointment at a private GP practice, which I do appreciate isn't cheap, but if it gets your teens the medications that they need, it's worth it. They might be more open to prescribing those meds. In my experience, the private GP appointment was totally unhurried, all the time in the world, and SO easy to book.
Your teens sound so unwell that I'd have thought it's worth a try.
The private GP then writes to your own GP letting them know what they've done, if you have one, and weren't effectively a visitor like I was.
Another avenue, along those lines, is to get a private appt at a Nuffield Hospital. (Or any other private hospital.) My dad's partner's wound was not healing, so we get her an appt at one, in desperation, with a private dermatologist at our nearest Nuffield hospital. That was a bit more expensive, at 220 pounds, but I think you will be able to see a psychiatrist for that cost, not just a GP. If you can afford it, I really think that would be the best thing. Again, the private doctor then writes to your own GP letting them know what they've done.
I know it's a lot of money, but if you can afford a few private appts, it might make all the difference. You might be all right with just one appt, if you get the meds your teens so desperately need, and then go back to your own GP. Becuase if they are working, your GP won't just take them off. If they don't work first time (as sometimes you need to try different ones) you might need more than one psychiatric private appt.
I am experienced with mental illness in the family, and I have seen the transformative effects of medication.
From what you've both described, your teens absolutely need to be on meds, and it's shocking that they are not, even after suicide attempts.
Edit: If this doesn't work or is not an option due to cost, you are simply going to have to INSIST on meds to your current GPs. Tell them that your child is suffering, the whole family is suffering, and that this simply cannot go on. And say that if your child kills themselves, having already tried, but were denied meds that could help, it'll be THEIR fault.
You can also put in a complaint, possibly to the GMC.
I cannot believe how shockingly bad it is not to give teens in this state of mind any meds.