Our earlier poll provides some ideas, but I want to supplement that with some narrative advice, of the kind I would give (and do give) to students.
Among the very top PhD programs in the Anglophone world, there are three viable choices for a student wanting to work on Nietzsche: New York University (with John Richardson and Tamsin Shaw), Princeton University (with Alexander Nehamas) and Stanford University (with Lanier Anderson and Nadeem Hussain). I am not sure how hospitable these places are for students primarily interested in Nietzsche, given the dominant interests of the faculty and most of the students, but they deserve serious attention from prospective students: you will get an excellent philosophical education and you have good philosophers who can serve as advisors with respect to Nietzsche work. NYU, with Beatrice Longuenesse, and Stanford, with Dagfinn Follesdal (part-time), Michael Friedman and Allen Wood, also offer good breadth of coverage in post-Kantian philosophy. Oxford University, another outstanding philosophy faculty, is also worth a look these days: Peter Kail, a leading Hume scholar, is working quite a bit now on Nietzsche. Again, there is the question about how hospitable the environment would be, but there are other Oxford faculty with sympathetic interests related to Nietzsche or other figures in post-Kantian German and French philosophy, like Michael Inwood, Stephen Mulhall, and Katherine Morris.
Among strong, but not very top, PhD programs there are several additional choices I would recommend: Birkbeck College and University College in the University of London system; Brown University; University of California, Riverside; University of Chicago; and University of Warwick. In terms of sheer numbers, and diversity of approaches to Nietzsche, Chicago has the most faculty to offer across various units, and for a student also interested in ancient philosophy and/or wanting wide coverage of 19th- and 20th-century European philosophy, Chicago has a great deal to offer. (Faculty interested in Nietzsche include James Conant and Michael Forster [Philosophy], Robert Gooding-Williams [Political Science], Brian Leiter and Martha Nussbaum [Law], Robert Pippin [Social Thought], and David Wellbery [German].)
Brown is stronger in most contemporary areas of philosophy (with a particularly good group in moral and political philosophy) than Chicago, but has less depth and breadth in post-Kantian philosophy of the 19th- and 20th-centuries. (The key faculty are Charles Larmore and Bernard Reginster.) University of California, Riverside also has a strong group in post-Kantian European philosophy, including Maudemarie Clark (a leading Nietzsche scholar, of course), Pierre Keller, Georgia Warnke, and Mark Wrathall, and UCR also offers solid, sometimes outstanding, coverage, across a range of contemporary areas of philosophical research, as well as in modern philosophy. University of Warwick has been a major up-and-coming department in the U.K. over the last decade, and is now solidly among the top ten U.K. programs. Keith Ansell-Pearson and Peter Poellner are the two main faculty interested in Nietzsche (their approaches are quite different, Poellner's being more likely to appeal to students with philosophy backgrounds), but other faculty do importnat work in Kant and post-Kantian philosophy (Quassim Cassam, Stephen Houlgate, A.D. Smith).
Birkbeck has my good friend Ken Gemes, a very talented philosopher who has supervised a number of students working on Nietzsche, and the Nietzsche scholar Simon May is also around and available to students. Birkbeck's main strength tend to be in contemporary areas of Anglophone philosophy--like philosophy of language, mind and action--but bear in mind that within the U of London system one can also draw on scholars like Sebastian Gardner, Mark Kalderon, and Thomas Stern, who are all interested in Nietzsche, making UCL another good choice.
Boston University, which has strong coverage of 19th-century philosophy, has just appointed Paul Katsfanas (whose Nietzsche work is known to readers of this blog) to a tenure-track position. BU thus deserves to be on the map for students thinking about graduate work on Nietzsche. Finally, University of Southampton, though not a strong department overall, is attractive for a student interested in Nietzsche, with Christopher Janaway and Aaron Ridley in Philosophy, and David Owen in Politics.