Post-graduate work appears to be apprentice based, since the elder Weasley went off to study dragons. I also got the impression that Aurors have additional training, but it is very likely in an FBI academy style enviroment.

Fudge's call to the PM was a courtousy call, from what I was able to gather. I don't think there's a formal connection between the two. As far as international government, there hasn't been a mention of an overarching UN like structure, but there do seem to be rather a lot of international guilds and associations.