A friend & I were in paris for new years 2001/2002 (after 9/11). her flight was at 9 a.m. CDG>LHR>LAX.
At about 4 a.m., she started packing and could not find her passport.
I called the US Embassy, (they seem to always have a person answer the line, was glad to be paying tax dollars for that). They put me in touch with some sort of consular rep., who interviewed my friend for 10 minutes (Where do you work? Where are your parents from? Do you have driver's license, etc.)
She apparently passed the test.
Consular said a letter would be faxed to BA advising BA to let her on the plane.
Indeed, when we checked in BA had a note, and let her check-in without passport. She transferred in LHR and presumably repeated her sob story in LAX, and was let in sans passport.

I think the rule is, if the airline flies someone to US who is not subsequently admitted (for example, b/c she doens't have passport), the airline is obliged to fly the person back to origin (though, without a passport, I don't know how my friend would've gottenback into the UK either).

Not a strategy I'd willingly try, but humans are humans and passports get lost, so it is possible to sometimes work things out. My friend had a compelling
reason she had to get back to US (some scheduled meeting with federal judge).

Good luck,
Shawn