There are flights from London to San Antonio TX and Albuquerque NM. Coool!
Actually, you will find most US cavers to be friendly and generous folk. Now, they might not take you to super secret Wonder Caverns your first day here, but we do try to show our guests a good time (though we we are prone to ask a zillion questions about your country,
whatever it is). Consider it one-to-one citizen diplomacy, to try to make up for the less than stellar
official attitude of the US government towards the world in general
-- as a people, we're not what you've seen on TV.
I think you'll like New Mexico, but being as this is a really big country, I concur with the others that two weeks is better than 5 days. One option might be to hook up with San Antonio cavers, then drive to Carlsbad, detouring to Caverns of Sonora (one of the best showcaves in the country) along the way.
Or figure out a circle of big caves to visit, (Sonora, Carlsbad, (is Kartchner open that time of year?) Cave of the Winds, pick one or two in Missouri, then go to Mammoth Cave in Kentucky, then Blanchard Springs in Arkansas, back through Austin, TX to San Antonio). Stay with cavers, and the expense is less than you think.
We've hosted a number of cavers through Missouri-- some Ukranians, including Dr. Klimchouk; Neil Kell from New South Wales--it's always been a grand time.