The Bible doesn't give any indication that there was any specific training or waiting period. However, it is likely that he had been receiving training prior to his father Amon's death and that such training continued for the first years of his reign. Also, it is very possible that his mother, Jedidah, acted as a co-regent with her young son during the first years of his kingship.
Josiah was a very different type of king than his father or grandfather, so it is possible that God blessed Josiah with unique maturity beyond his age when he ascended to the throne. This may have allowed him to become respected as king despite his youth.
Amon...did evil in the eyes of the LORD, as his father Manasseh had done...Josiah...did what was right in the eyes of the LORD and walked in all the ways of his father David, not turning aside to the right or to the left. (2 Kings 21:19-22:2)