The NHS is being heavily charged by nearly every pharmacy company that it has to buy from, with many of them foreign companies.
I would see no problem inviting American companies to compete on the production and delivery of some medicines, or even equipment. We know they are excellent businessmen, and competition on this front shouldn't do any harm, if managed properly.
My sister lives in Germany, and their system is quite expensive, but there again she wants for nothing, literally. She has regular, paid for, checkups for any and all conditions, including hospital visits at a days notice, along with any medication.
Nat has a good point in that if the NHS was managed differently, with those in need coming first, and those who would like, being supported secondly.
Involving the Americans doesn't mean that the NHS would change for the worse overnight. As for private insurance, etc, we have this already, with the likes of Nuffield and Benenden, who provide a service that many people already make use of. (I bet I'm not the only one on here who has made use of their facilities).
The NHS will remain the same in principle, but there is no reason whey it has to be used in such a haphazard way. These last few years I have been extremely grateful to the NHS, which has, without doubt, saved my life. Such availability should remain for everyone and if we have to tweak the system to ensure that, then so be it.