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]

As another point of comparison, here are some counts from the [Corpus of Contemporary American English](https://www.english-corpora.org/coca/) (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](https://english.stackexchange.com/questions/583564/is-nor-used-correctly-here-as-a-conjunction-nor-at-the-beginning-of-a-sent) about whether "nor" can be used at the beginning of a sentence. COCA indicates it often is.)



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