Can I get a British emergency passport?

HELP!
Right, I am a 15 year old British citizen who lives abroad.
My parents, at the start of May, sent my passport away to Dusseldorf, Germany to get renewed and it got lost and no one knows where it is.
Luckily we took note of my passport details (number, expiration date etc.)
But because there is starting to be a change in the passport system, a passport renewal will take at least 8 weeks (there is a backlog of passports).
This is a huge problem for me because in August (3rd-28th) I am travelling (alone) to America to see a friend of mine who is dying of cancer.
Am I able to get an emergency passport? And if so, how long does it last for?
What should I do? :( My friend doesn’t have long left and I really want to spend time with her before she passes.
Any help?