Upvote:5
So you have three problems:
Getting back into the US is certainly doable. Green card holders do NOT need to show a passport to return to the US - just a Green Card will suffice.
This then leaves us with the UK. As a British citizen, according to info from the Passport Office:
A person who is a British citizen is not subject to immigration control and is free to enter or leave the United Kingdom without restriction. A British citizen who travels on a passport issued by another country will need to apply for a Certificate of Entitlement to the Right of Abode to be endorsed in his passport to confirm he has unrestricted entry to this country.
So in theory, 'free to enter or leave' - you would arguably be allowed in. Of course you'll want to be able to prove who you are, and that you're British, and be very aware that you'll likely get questioned about doing this.
Finally, the airlines. This might be your biggest problem. Each airline (and indeed, airline employee) is likely to be different in their handling of this. It's extremely likely that they'll have concern over a person travelling without a passport or ETD - they want to know you're legally allowed into the country they're taking you to, or they can be fined and made to pay to remove you. So if there's any doubt over the matter, they may just say no.
So your steps:
Alternatively, look into a second passport, or an ETD (Emergency Travel Document).