It's a really tough situation you're in.
Would you consider taking matters about your nephew's mental health into your own hands whilst he's under your care?
You could take him as a visitor to your own GP but it would be more ideal if he could see someone from his own practice - if practical. Although, you could phone his practice and ask for the GP to call back.
Would you be willing to engage with the school and explain the situation you've been left in - not dropping your sister in the shit, as such, but they may be able to offer some guidance, experience, knowledge to help you.
My son was diagnosed with depression when he was 15. It wasn't a surprise and we had been on the path to get help (since he was 8) but it wasn't soon enough and he ended up in hospital after a suicide attempt. The scariest part was how quickly his mood turned to spiral down enough for him to try.
Unfortunately, we have had to go down the private route as CAMHS simply couldn't act fast enough. Actually they made things far worse initially by telling him he was autistic. He isn't. He does have some traits of Asperger's (like DH and DD2) but not to cause this.
Keep talking. Phone Samaritans if you're finding it overwhelming - they will offer support but not advice. They may be able to signpost you to other organisations that may help.
Ultimately, your sister needs to take responsibility and not assume he's a "teenager being attention-seeking". She ought to be validating his feelings and helping him not berating him.