Assuming I got a PhD in Computer Science from Oxford or Cambridge University, how would that affect my chances at becoming a professor at a top US university versus getting a PhD from a top US institution? Assuming also that I had US citizenship?
I am an American who earned a Ph.D. at Cambridge University. A degree from Cambridge or Oxford is well-respected in the US - I was warned to avoid other UK universities as they are not as well known in the States. The one caveat is that all of your network for post-docs and jobs will be in the UK, not in the US. It was an amazing experience though so I would not want to discourage you!
In mathematics, I would say that British Ph.D.s are very well respected in the U.S.A. Probably more so than many US universities. The US citizenship is a plus, because it means that the university can avoid the hassle of H1 visas and Green Cards. But a major research university will probably not reject you if you don't have US Citizenship.
My guess is that it the effect on academic employment would be relatively small compared to other factors, as pedigree is typically neither necessary nor sufficient for obtaining employment. Arguably, department reputation will matter more, but what you do while you are there will matter far more than that.
It might be helpful to your decision to note that several UK universities are very well respected in the US and elsewhere: Imperial, UCL, KCL, Edinburgh, Manchester, Sheffield, to name a few. Are Oxford and Cambridge the only UK institutions with respected CS departments?