How Long to Fully Complete World Transfer Service?

So I decided to move one of my ancient starter characters on Coeurl over to Balmung and remake him with both a new look and a new name. The transfer went through fine: the Mog Station and Lodestone both list him as being on Balmung now, and I can log in and play him on Balmung. But whenever I try to use the name change service, it says the transfer process is still in progress, and I have to wait for it to complete.

What exactly still needs to be done? Should I just hang out and wait to get an email saying the transfer is completed?

Mine took seconds... and that was even back when the service first started. Don't know what the holdup is since I've never used the name change service.

I think the name change thing takes a full day so I would wait the full 24 hours just to be safe. If that doesn't work, contact SE support.