What did this look like for you? What should I ask of them?
My experience dealt just with my and my son's US passport renewals. It was about 10 years ago when US law changed so that US citizens needed passports for trips that previously didn't require them -- short trips to Mexico and the Carribean for example. We had applied to renew our passports months before a planned trip to the UK but because of a surge in passport applications, the passport agency got way behind.
After many contacts with the agency and standing in line for hours at the Los Angeles federal building, we kept getting told that our renewals were being processed and that there was nothing to do to speed things up. Eventually, since our trip was a few days away, we contacted the office of Henry Waxman, our representative. I believe the staff member there had us fax him our travel itinerary, names, etc. The next day we picked up our renewed passports at the federal building.
So I'm not sure if your representative's office can do anything about the visa itself, but where they could really help is if the US passport agency isn't reasonable about issuing you a replacement immediately. As I said, hopefully you won't need their help, but if you do, I would let your congressperson's office know that your passport was lost by UKVI, that the loss of the passport is creating a hardship for you, and that they can help by speeding up the replacement.
Keep us updated. Sending good thoughts your way!