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, which shows that "but neither" is much more common than "but nor" (although the latter seems to be increasing recently):
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 ...