A bit vague
It's very difficult to say. It all depends on what you're on, what you're moving to, what you're migrating and how you're migrating. :)
How big are inboxes?
Is the new mail system on the same network? What bandwidth is available? Is it feasible to use disk-shipping if it's a remote system?
What storage backend does each system use?
If they're both maildir then the task is quite simple - copy all the files to the new system and perhaps script a few tweaks to some of the files.
If you use something like IMAPSync then it's going to take quite a bit longer. A large mailbox over the internet might take hours.
Is it just mail too, or are you going to try and migrate contacts and even calendars? Unless both systems use some kind of standard (CalDAV etc) then it's going to be a manual process - export, convert, import. Even with CalDAV etc. it's going to be fiddly.
So without having anything to go on, I'd say it'll take a few weeks, if not more.
Biting the hand that feeds IT
