We have a few issues going on with character transfers right now. Listed here.
It sounds like the character may have something preventing it from transferring, and the error message you are supposed to get, isn’t displaying.
Check your characters for Heirlooms, Mail, active Auctions, and caged pets.
If you can tell me which character you are attempting to transfer I might be able to take a look if you can’t immediately figure out what may be causing the issue.