Well, it may be immediately to you as the coach who cut the kid (and I'll take your word that this is the case as it's been a while since I've cut a player), but to everyone else, the kid doesn't show as available until the start of the next cycle.
For example, all the kids that get cut in the cycle immediately prior to signings don't show up to anybody else until the start of the signings cycle. Plus, I know I've checked the "News" tab of a given player's former team after he was cut and often it shows it was in the prior cycle, but I know he wasn't recruitable in that same cycle.
So, from a practical standpoint, it's at the start of the next cycle although his former coach may be able to see him as immediately available.