The recommendations are premised on three assumptions about what is needed to do good PhD work on Nietzsche: (1) a strong, general philosophical education; (2) good Nietzsche scholars to supervise the work; and (3) a philosophical environment in which one can get a solid grounding in the history of philosophy, especially ancient philosophy, Kant, and post-Kantian German philosophy.
With that in mind, here's the eight programs I'd strongly recommend for someone certain they plan to focus on Nietzsche:
Birkbeck College, University of London: a solid department overall, albeit a bit narrow (top 15ish in the UK), unusual in having two Nietzsche scholars on faculty: at the senior ranks, Ken Gemes [who is only part-time] and at the junior ranks, Timothy Stoll. If one reaches out to faculty at other London colleges, one can also get the necessary historical education in other figures.
Brown University: a strong department overall (top 20 in the US), with one leading Nietzsche specialist, Bernard Reginster, and two other senior faculty with sympathetic and complementary interests (Paul Guyer and Charles Larmore). Guyer and Larmore, as well as Mary Louise Gill, provide strong coverage of other important periods and figures for purposes of studying Nietzsche.
Columbia University: a very strong department overall (top 10ish in the US), with three senior faculty with interests in Nietzsche (although only one is a specialist, Gooding-Williams): Taylor Carman, Robert Gooding-Williams, and Frederick Neuhouser. With these three, as well as Lydia Goehr and (part-time) Axel Honneth, also one of the best places in the U.S. to study the Continental traditions in philosophy. Also offers strong coverage of ancient philosophy and Kant.
New York University: the best department in the Anglophone world, with three senior faculty with interests in Nietzsche: Robert Hopkins, John Richardson, and Tamsin Shaw (though only Richardson seems to be actively working on Nietzsche these days). The department now also has strong coverage of ancient philosophy and through Richardson and Anja Jauernig solid coverage of Kant and the post-Kantian Continental traditions. Given the department's dominant strengths in other areas to date (e.g., metaphysics, philosophy of mind), so far there have been hardly any students there working on Nietzsche or other post-Kantian figures--something a prospective student should investigate.
Oxford University: a very strong faculty (top 2-3 in the Anglophone world), with strong coverage of the history of philosophy, with one significant senior Nietzsche scholar (Peter Kail) and one strong younger Nietzsche specialist (Alexander Prescott-Couch). Stephen Mulhall, Joseph Schear and Mark Wrathall offer good coverage of other aspects of the post-Kantian Continental traditions, especially Heidegger and phenomenology. Also outstanding in ancient philosophy.
University of California, Riverside: a solid department overall (top 30ish in the US) and one of the best places in the U.S. to study the Continental traditions in philosophy with two important senior faculty--Maudemarie Clark (a leading Nietzsche specialist) and Pierre Keller (Kant, German Idealism, phenomenology)--as well as the recently tenured Sasha Newton (Kant, German Idealism) and Georgia Warnke in Political Science. The department is especially notable for the way in which the study of the Continental traditions is closely integrated with the study of the rest of philosophy, to the enrichment of both. (It's also a very collegial place, one of my favorite departments to visit in the country.) There is also a large and impressive group of graduate students working on the post-Kantian traditions and/or interested in Nietzsche.
University of Chicago: a strong, if somewhat idiosyncratic, department (top 20ish in the US), with particular strengths in ancient Greek and Roman philosophy and in Kant and post-Kantian German philosophy. Chicago has to have more scholars interested in Nietzsche from more divergent points of view than anywhere else: besides me, also Martha Nussbaum, Robert Pippin, David Wellbery, and (part-time still) James Conant and Michael Forster. As with Riverside, there is a large group of students interested in Nietzsche (six of the eight PhDs I've worked closely with in recent years have had serious Nietzsche interests, two have published on Nietzsche, and one wrote a dissertation with a significant Nietzsche component). (Note: All of Pippin's supervision in German philosophy in recent years has been of students working on Kant or Hegel.)
University of Warwick: a solid department overall (top 10ish in the UK), with one notable Nietzsche scholar (Andrew Huddleston) and strong coverage generally of Kant and the post-Kantian Continental traditions (e.g., Quassim Cassam, Stephen Houlgate [who also is interested in Nietzsche]).
Here are some other departments a student interested in Nietzsche should certainly consider as well, although they are not as strong as the preceding in my judgment:
Boston University: a solid department (top 50ish in the US), with a strong commitment to the history of philosophy, including Kant and the post-Kantian Continental traditions (e.g., Daniel Dahlstrom, Sally Sedgwick). BU has one well-known Nietzsche specialist (Paul Katsafanas, though he is pushing a rather distinctive, and to my mind, implausible line about Nietzsche these days, though I still highly commend several of his earlier papers that we've discussed on this blog in the past--but students sympatico to his approach would no doubt find him an excellent person with whom to work).
Stanford University: a very strong department (top 10ish in the US), with two senior faculty who have done important work on Nietzsche: Lanier Anderson and Nadeem Hussain. In the past, I would have put Stanford in the top group, but Nadeem tells me he's not really working much on Nietzsche anymore. Also strong in ancient philosophy and, with Anderson and Michael Friedman, also very good for Kant. The department's center of gravity, judging from its PhD graduates, does appear to be more in logic, language, mind, metaphysics and epistemology.
University College London: a good department (top 10 in the UK), with two faculty who publish on Nietzsche: Sebastian Gardner and Tom Stern. Gardner is also a major scholar of Kant and German Idealism. Gardner is excellent, Stern's work is weak.
University of Essex: a narrow department, but strongly focused on Kant and the post-Kantian Continental traditions. One well-known Nietzsche specialist on faculty: Beatrice Han-Pile.
University of Southampton: A solid but not top 15 UK department, with a particular strength in Schopenhauer and Nietzsche--most notably Christopher Janaway, but others in philosophy or cognate units include David Owen, Aaron Ridley, and Tracy Strong. Note that Strong is in his mid-70s.
For a student looking to do a terminal M.A. first, s/he might consider any of the UK departments (where students first do a master's degree or B.Phil. before doing the PhD), or, in the U.S., Georgia State University remains far and away the best choice: in addition to solid coverage of moral, political and legal philosophy, ancient philosophy, and philosophy of mind and cognitive science, the department has two well-known scholars who work on Nietzsche (Jessica Berry and Gregory Moore), and two other faculty who work on Kant and post-Kantian German philosophy (Sebastian Rand and Eric Wilson).
The best Nietzsche scholar on the European Continent is Mattia Riccardi, now at the University of Porto in Portugal. Also in Portugal, The New University of Lisbon continues to have a lively philosophical community interested in Nietzsche led by Joao Constancio. Andreas Urs Summer at the University of Freiburg in Germany is doing interesting historical and philological work, albeit of somewhat less clear philosophical import.