Upvote:6
You would most likely have to get a C Visa in London.
https://uk.usembassy.gov/visas/transiting-the-united-states/
Processing times are 3-5 work days and it's not cheap either https://uk.usembassy.gov/visas/transiting-the-united-states/processing-times-return-of-passport/
You may get lucky by showing up there in person and making a case for your special circumstance. However this is absolutely unpredictable. We had multiple experiences where consulate employees made up rules, timelines and requirements on the fly, regardless of what the official rules were, what your appointment letter stated, or what you had previously confirmed on the phone (multiple times!).
It would be far easier to fly directly to Mexico on British Airways or Aeromexico or route through Europe (Amsterdam, Paris, Frankfurt). I checked flight costs and they were all about the same, about $1500 round trip for departure tomorrow and return on the weekend. Another option could be Avianca through Bogota.