The system will consider a question as "answered" after one of the answers has at least 2one up votesvote. While it won't reward you rep as "the accepted answer", or mark it as accepted, it will remove the question from the unanswered
queue.
As for nudging the user, there is nothing wrong with leaving a comment on the question requesting the author select an answer - or improve their original question if it's still unanswered.
It's hard to know if the solutions provided are actually viable fixes for this situation or if the user just is unaware of how the system works, so make sure to take that into consideration when gently nudging users to accept answers :)