Dude you lost your passport? You aren't in the system now, you're basically a Jason Bourne and will be hunted down by the government. That or you could just go get a new passport at your Embassy, or if that's too far I would guess they have travel dates. It shouldn't affect your AOS, it could slow it down if it is required at some point but you will just need to get one. Odds are you will get your green card in the mail and nobody will know about the passport. I would work on getting your replacement. Unless your address changed, no need to report to USCIS. Good luck.
You need to report it lost to the UK https://www.gov.uk/report-a-lost-or-stolen-passport
You need to order a new one from the UK. They are not ordered from British embassies in the US. https://www.gov.uk/overseas-passports
You will not be in trouble with US immigration. Save documentation that you did report it lost.
