I'm having the exact same problem with identical symptoms. The upload gets shown at the bottom of the main window, but the Progress and Last Updated fields never change. I've tried waiting, canceling and re-starting, uploading multiple files, small/large files, etc. Nothing seems to work.
I'm using a 10.9.5 and have tried Chrome v.39.0.2171.95 (64-bit), Opera v.26.0.1656.60, and Safari 7.1.2. I'm using vSphere v.5.5.0 Build 2175560 and Client Support version 5.5.0 Build 1896823. All three browsers have the same behavior and none work.