I'd agree with @JohnLawler's comment

>  *But neither* is the normal use; *but nor* strikes me as novel, though I don't really know. 

This is supported by a query to the [Google ngram viewer](https://books.google.com/ngrams/graph?content=but+neither%2C+but+nor&year_start=1800&year_end=2019&corpus=26&smoothing=3&direct_url=t1%3B%2Cbut%20neither%3B%2Cc0%3B.t1%3B%2Cbut%20nor%3B%2Cc0#t1%3B%2Cbut%20neither%3B%2Cc0%3B.t1%3B%2Cbut%20nor%3B%2Cc0), which shows that "but neither" is much more common than "but nor" (although the latter seems to be increasing recently):

[![enter image description here][1]][1]

It would be nice to do a corpus search on "[comma] neither" and "[comma] nor" for comparison. In fact I'd like to quantify the relativ frequency of the eight constructions `{comma, semicolon, period, "but"} × {neither, nor}` — if anyone knows how to do that ...



  [1]: https://i.sstatic.net/pgtFk.png