It pretty much is instant - I transferred from my original 1.0 server (Ridill, a JP server) to Balmung (US server), and once the payment was successful, I simply logged into the game, and my character data showed up in the new server group pretty much straight away.

As for transferring to a 'EU' server, it should be noted this is actually a misnomer, as there are no true 'EU' servers at this time - the servers stated as being 'EU' are still actually part of the NA server groups and are physically located in the same place (Montreal Canada).

The only benefit for choosing a 'EU' designated server is they are a place where German and French speakers are more likely to be found if you happen to have that version of the client. I'm not sure about server time though, as from what I understand even the NA servers are timed to the JP server group.