If I remeber correctly you shouldnt need to login to the Sky Go website just to play existing downloads, you should just be able to login directly within the app and it will work. Maybe try closing the browser and app, then launch the Sky Go Player app on it's own. Does it then let you access existing downloads? Does it ask you to sign in?
The existing Sky Go Silverlight implementation on Windows and the Cisco interim implementation on Mac will be replaced in the future by one new solution. This should deliver a much better and consistant Sky Go user experience across the platforms.