This is an intriguing thread as I am a U.S. student who is in the process of applying for a postgraduate course in the U.K. (History). I have been researching U.K. universities for the past year and have a few comments regarding the reputation of top US/UK universities in their opposing geographies.
General US opinion of UK universities: In general, the Oxbridge universities are clearly the most prominently known among most Americans (at least among those who have not looked beyond the mass media). In my experience, most Americans, even those in academia, are not familar with the second and subsequent tiers of UK universities.
General UK opinion of US universities (based on my reading and exchange with UK students): The Ivy League (Harvard, Princeton, Yale, Columbia, University of Pennsylvania, Dartmouth, BROWN, Cornell), the University of California schools (particulary Berkeley, UCLA and UC San Diego), MIT, Caltech, and Stanford, are highly reputable. I could probably include the University of Chicago and Duke.
The US holds the numeric edge due to the vast number of schools, funding
(this attracts prominent researchers), and publicity (especially among the private universities).
My personal opinion: Both countries offer an excellent education. It saddens me that most in the US aren't familar with the other great universities outside Oxbridge (many U of London schools, LSE, Bristol, Durham, Warwick, etc...).
My reasons for pursuing postgraduate work in the UK? The flexibility (particulary in the Humanities) of designing your own research focus at the MA/MRes level appears to be strong in the UK. US undergraduate work is less focused but more comprehensive (liberal arts) but most MA programs are more compartmentalized. In my opinion, the UK system (undergraduate and MA/MRes) is geared more toward pure focused research and therefore a better prepratory program for PhD/DPhil.
Just a humble American.