You should always start your own thread when your issue.

Is this occurring for one use on one machine or multiple  users across multiple
machines?

Multiple users and multiple  machines, it doesn't work
For the widget for drag and drop looks like there needs to be Office 2010
installed on the client but the old 2007 upload  should have worked.

As per the documentation on technet if I understand correctly It won't work
unless Office 2010 provide a Firefox control adaptor



technet.microsoft.com/en-us/library/cc288142.aspx







File upload and copy


Requires an ActiveX control, such as the one that is delivered in Microsoft
Office 2010, and a Firefox control adaptor. Microsoft Office 2010 does not
provide a Firefox control adaptor for this control.

That's a known limitation -
technet.microsoft.com/en-us/library/cc263526.aspx