There are sometimes cheap fares for London leaving from NYC. You might be able to find a cheaper total price if you search as two different fares – one from texas to NYC, and one from NYC to London. It’s worth a shot. The best site for searching flights is http://kayak.com
But it does sound like $1200 r/t for a flight that leaves this Wednesday is a pretty good deal. I bet you could find something closer to $800 if you were able to have a 14-day advance purchase, but it sounds like you need to get there sooner.
Do you fly a lot within the US – and do you have a frequent flier account for any domestic airlines? If so, many domestic airlines are part of international alliances that would fly to London – so if you are going to call and try to get a special rate for sick family members, I would first try calling the airline that you have the best relationship with (the most frequent flier miles) – start by telling them that you have always been a loyal customer, etc, and then explain your plight. They may be more inclined to help you if it is a customer loyalty thing.
I am not sure if http://priceline.com will allow you to do the “name your price” thing for an international flight, but you might want to check that out. You can say how much you’d be willing to pay for a ticket, and if they can match that price they buy you the ticket. BUT you don’t get to name your departure time (just date) and you might get stuck with a random stopover somewhere. However, if you’re desperate to find a cheap flight, and you have some flexibility with what time you leave/arrive, it seems like this might be an option.
p.s. to go to London I don’t think you need any other paperwork besides a passport. As long as you have a valid US passport you should be OK.