How old does an infant to be to be moved in to there own room and not share it with a mother and her boyfriend?

Whenever you start feeling uncomfortable with the situation. Many cultures room in together.

There is no allowed it is up to it is use your own judgment/preference.
My grand baby has been in his own room since birth. As long as their crying in the middle of the night will wake you up and you feel comfortable with them in their own room then you can do it.