It’s immediate, or as slow as the number of toons transferring. There’s no wait to transfer the next one.
One toon of mine was transferred after 5 minutes, another was ready next morning.
This varies depending on circumstances and how blizzard decide to deal with it. It is most often an alleviation to migrate off of near dead servers. I asked for the current Hydraxian Waterlords > Pyrewood transfer.
Or well it’s been a while since I checked and with my luck the free transfer has ended before I decide to use it. It better not, I paid my sub for it and are just slowly making sure I just don’t abruptly leave behind good friends and a well organized account.