There is a simlink from the modules base name to the current version of the module, so if that's not there after your move, it could cause issues.
eg the core module is jrCore and is located at:
/modules/jrCore
But the first update that comes in from the marketplace is placed with its version number in the folder title, like this:
/modules/jrCore-release-5.2.16
/modules/jrCore-release-5.2.17
/modules/jrCore-release-5.2.18
and the /modules/jrCore becomes a simlink to the most recent version.
What I would probably do if I couldnt copy the simlink would be to put the most recent version of the modules back on their base url, so rename
/modules/jrCore-release-5.2.18 to
/modules/jrCore
and only upload those base versions to the new server, then let the marketplace take care of making its own simlink on the server when updates are done.
--edit--
I'm not sure if you're meaning you cant use the site, or just that you cant update from the marketplace. If its just that you cant update from the marketplace, it might be because the web user does not have permissions to change the simlink thats been uploaded via FTP.
updated by @michael: 01/10/16 10:30:17PM