This use of neither is at best marginally grammatical. It is certainly extremely rare: all the examples of Neither do/does that I looked at in GlowBE (the Corpus of Global Web English) are "afterthought" - following a negative statement
He doesn't want to. Neither do I.
rather than "forethought"
? Neither does he want to nor do I.
(It's not even clear to me whether to invert the subject and verb in this construction or not: ?Neither he wants to nor do I.)
With other coordinates (noun phrases, prepositional phrases, verbs, or verb phrases) it is fine:
I want neither tea nor coffee.
I do it neither at home nor at work.
I neither wanted nor asked for this.
I neither rang the bell nor waited for them.
It's just full sentences where it doesn't occur. This is also true for the other forethought connector, both ... and.
I both rang the bell and waited.
but probably not
? Both did I ring the bell and I waited.