Are you able to view other Flash content on the same Mac in Safari? This sounds like a Flash install problem rather than a page-specific problem, unless all other Flash works fine on the browser.
Plugins won’t play at all when previewing in Safari. The only thing I can think of that might have caused this change is the recent sandboxing of plugins in that particular browser. Everything should work as expected when the site has been uploaded (do let us know if this isn’t the case).