Just wondering, because I can't seem to find any...I'm particularly interested for physics/natural sciences
Cambridge do a year in MIT
There are definitely no exchange programs with Harvard and Princeton - not sure about Stanford. Maybe Imperial does something with Stanford but I doubt it. MIT and Cambridge do collaborate, yes.
caltech and corpus christi cambridge
Oxbridge will be the only ones, our other universities aren't well known enough to compete with the top US universities you've mentioned.
Imperial has exchange links with Columbia, Carnegie Mellon (ChemEng) and MIT (Material).
I heard that UCL has links with UChicago (Econ). CalTech and UPenn, can't remember which subject
Pretty sure LSE has exchange links with Columbia
Stanford Mathmos can do exchange trips with both Japan (Kyoto) and France (Paris) in one course ie do both.
University of East Kensington do an exchange course with Harvard.
UCL for law does a thing with Columbia.