When did Muhammad meet the Ancient Israelites?

Answer 1

Muhammad never met the Ancient Israelites, although he met their descendants, the Jews of Arabia, on numerous occasions, both in Mecca, where he would see Jews incidentally, and in Medina, where originally half of the city's population. He learned about the Ancient Israelites through his interaction with the Jews of Arabia and the Christians of Arabia, as well as, according to Muslims, his revelations from God.

Answer 2

Your question conflates two different historical periods. By the sixth century CE, we had for many centuries no longer been called Israelites. Rather, we were called Jews.
Muhammad lived in proximity to large numbers of Jews and met them in his daily business many times, from his youth.