The last version was 2012, so it warrants some updating given changes in the interim (and also some of the useful comments on the last version). 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 (certainly top 10 in the UK), unusual in having two very substantial Nietzsche scholars on faculty, Ken Gemes and Andrew Huddleston. 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 interests in Nietzsche (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 interested in Nietzsche: 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, now with three senior faculty with serious interests in Nietzsche: Robert Hopkins, John Richardson, and Tamsin Shaw. The department now also has strong coverage of ancient philosophy and through Richardson, Anja Jauernig and Beatrice Longuenesse, has strong 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 few students there working on Nietzsche or other post-Kantian figures--something a prospective student should investigate.
Princeton University: a very strong department overall (top 5ish in the US), with one leading figure in Nietzsche studies, Alexander Nehamas, who has returned in recent years to working on Nietzsche and supervising students (e.g., Huddleston at Birbeck, above). Also very strong in ancient philosophy, with other faculty in Philosophy or cognate departments offering coverage of Kant and post-Kantian German philosophy (mostly 19th-century).
University of California, Riverside: a solid department overall (top 30ish in the US) and one of the best places in the U.S. (perhaps the best) to study the Continental traditions in philosophy with Maudemarie Clark (a leading Nietzsche specialist), Pierre Keller, and Mark Wrathall, as well as Georgia Warnke in Political Science and a new junior faculty member in Philosophy, Andreja Novakovic,. 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 and French philosophy. Chicago has to have more scholars interested in Nietzsche from more divergent points of view than anywhere else: besides me, also James Conant, Robert Pippin, David Wellbery, and (part-time still) Michael Forster. As with Riverside, there is a large group of students interested in Nietzsche (four of the six PhD students I'm currently working fairly closely with have substantial interests in Nietzsche, though most are not writing dissertations in German philosophy).
University of Warwick: a solid department overall (top 10 in the UK), with two senior scholars interested in Nietzsche (Keith Ansell-Pearson, Peter Poellner) from different perspectives, and strong coverage generally of Kant and the post-Kantian Continental traditions (Quassim Cassam, Stephen Houlgate [who also is interested in Nietzsche], and A.D. Smith, among others).
Here are some other departments a student interested in Nietzsche should certainly consider as well:
Boston University: a solid department (top 50 in the US), with a strong commitment to the history of philosophy, including Kant and the post-Kantian Continental traditions. The Nietzsche scholar Paul Katsafanas was recently tenured there (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).
Oxford University: a very strong department (top 5 in the Anglophone world), with strong coverage of ancient philosophy and the history of philosophy, but only one significant Nietzsche scholar on faculty, Peter Kail. Stephen Mulhall and Joseph Schear offer good coverage of other aspects of the post-Kantian Continental traditions.
Stanford University: a very strong department (top 10 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 & epistemology.
University of California, San Diego: a strong department (top 20ish in the US), with two senior faculty interested in Nietzsche (Michael Hardimon and Donald Rutherford), and extensive coverage of ancient philosophy and Kant.
University College London: a good department (top 5 in the UK), with three faculty with interests in Nietzsche: Sebastian Gardner, Mark Kalderon, and Tom Stern--though for none does it appear to be a primary interest, except perhaps Stern (though I have mixed views of his work). Gardner is also a major scholar of Kant and German Idealism.
University of Essex: a narrow department, but strongly focused on Kant and the post-Kantian Continental traditions. Two faculty do notable work on Nietzsche: Beatrice Han-Pile and David McNeill.
University of Southampton: another narrow department, but with a particular focus on Schopenhauer and Nietzsche--notable faculty include Christopher Janaway, David Owen, and Aaron Ridley.
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).
On the European Continent, the place to be for someone interested in Nietzsche now is the University of Bonn, with Michael Forster, a preeminent scholar of German philosophy of the 18th- and 19th-centuries, as well as ancient philosophy, and Mattia Riccardi, the best younger Nietzsche scholar in Europe in my experience (he also works on Kant and philosophy of mind and cognitive science). The New University of Lisbon continues to have a lively philosophical community interested in Nietzsche led by Joao Constancio.