Well, here's what it looks like I have to do:
1) My company is cancelling my FLR application, as the urgent request still will not get the passport back to me (alnog with my work permit which has been approved) on time.
2) When I get to America, I will have to make an appointment with the UK embassy and get "Entry Clearance". The woman from UK Immigration said that this should not be a problem since I alreayd have the work permit.
3) When I get bck to the UK , my company will need to re-file my FLR application. It will take at least 10 weeks to then get my passport back, but that shouldn't be an issue since I have no travel plans in the near future.
So, the only downside for me is that we will have to spend one day of our holiday in NY getting the Entry clearance sorted out, which isn't the worst thing in the world since I was planning on taking my partner there for the dy anyway. Hopefully, all of the information I have been given is correct, otherwise, I may be stuck in the states for a while.
Thanks everybody for your help, and cross your fingers that the passport makes it back to me on time!!