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):

As another point of comparison, here are some counts from the Corpus of Contemporary American English (COCA):

• NEITHER: 74071, . NEITHER: 13370, , NEITHER: 9902, BUT NEITHER 3732, ; NEITHER: 716
• NOR: 88449, , NOR: 27236, . NOR: 11475, ; NOR: 1935, BUT NOR: 118

This supports the idea that "but neither" is somewhat rare but "but nor" is much rarer. (It also speaks to your other question about whether "nor" can be used at the beginning of a sentence. COCA indicates it often is.)

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):

As another point of comparison, here are some counts from the Corpus of Contemporary American English (COCA):

•  

  • NEITHER: 74071, . NEITHER: 13370, , NEITHER: 9902, BUT NEITHER 3732, ; NEITHER: 716
  • NOR: 88449, , NOR: 27236, . NOR: 11475, ; NOR: 1935, BUT NOR: 118

• This supports the idea that "but neither" is somewhat rare but "but nor" is much rarer.

• it supports the idea that , neither is common.

• It also speaks to your other question about whether "nor" can be used at the beginning of a sentence. COCA indicates it often is.

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 ...

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):

As another point of comparison, here are some counts from the Corpus of Contemporary American English (COCA):

• NEITHER: 74071, . NEITHER: 13370, , NEITHER: 9902, BUT NEITHER 3732, ; NEITHER: 716
• NOR: 88449, , NOR: 27236, . NOR: 11475, ; NOR: 1935, BUT NOR: 118

This supports the idea that "but neither" is somewhat rare but "but nor" is much rarer. (It also speaks to your other question about whether "nor" can be used at the beginning of a sentence. COCA indicates it often is.)