Existing Cascaded Lookup will be out of work in a new site which is created by site template, the values in those columns will be gone. But the newly created Cascaded Lookup columns in the new site will be working without any problem.
No solution for existing Cascaded Lookup columns.
Existing Cascaded Lookup column can’t adapt to a new site which is created by a site template, since the web ID, list ID and view ID are totally changed after the recreation.
Applies for: SharePoint 2013, SharePoint 2010, SharePoint 2007