Did not change address when moving into NYC

I lived on Long Island (Suffolk County) and worked on Long Island as well. I moved into NYC (Queens) in June and updated my address with the state and on other things but failed to inform my employer and update my address on my taxes. Since the business is on Long Island, I did not have NYC taxes withheld for this past half year. Should I just wait until the new year in a week to update my address and start having NYC taxes withheld for the next tax year? When doing my taxes for this past year will I have to pay the NYC tax that was not withheld?