I think the point of the 620 film size, and its larger brother 616, was to use slimmer spools so that the cameras could be made a bit slimmer. As part of this, the spools were made of metal from the outset (back in the early 1930's). I have never seen or heard of a 620 spool with a wooden core.
For decades 120 spools had metal ends and wooden cores but from around the 50's onwards most manufacturers were using all metal spools. I accept that Agfa apparently was an exception. I think they must have changed in the early 60's as the Agfa spools I've seen from around then were metal (they had their name engraved in the ends). Later everyone seems to have changed to all plastic spools for 120.