...nor he heard anything
Here is a page with several examples of "Neither...nor...", including
"
Neither did he take his profession seriously,
nor considered enhancing his skills to do well in his career."
To my ear, this also sounds wrong.
In my opinion it could be made grammatical (but still literary and old-fashioned) by saying "...nor did he consider..."
On the same page we have—also literary and old-fashioned—
"
Neither had a safety-cum-risk assessment been undertaken[,]
nor had the authority granted product approval."
Your sentence could become "Neither did he see anybody
nor did he hear anything. "
P.S.: On the same page, similarly with inversion of the auxiliary verb, we have
"Neither was his case politically motivated nor the party leader a victim of political vendetta."
In my mind I can make this grammatically acceptable only by inserting "was": "...nor
was the party leader..."