What would happen if every bank in the world were replaced by a single institution requiring handwritten signatures and banning online banking?

Since such a singularity is not likely to happen. No one knows for sure. Would there be a universal currency all around for the world? The bank would probably be spending a serious amount of resources, just doing currency conversions that it sets for its cross-border or cross-currency transactions. Traffic would be a bitch always around and/or near bank branches for sure.

You'd have some parallel form of banking or trading emerge as a result of the ban on online banking for sure. People will look at convenience, and in the absence of it, they will find a social hack for it.