http://www.disney.co.uk/toontown/contactus.html
If you are lucky they will reply over the weekend; if you highlight it as a billing enquiry you might even get it dealt with as a priority!
Or...from clicking about...I think the way to do it is...
Get to the screen under HELP -> Manage my Account
Make sure you add the new card as a "payment tool"
Then go to "my subscriptions"
and use "Change payment tool for this subscription to:" to pick your new "payment tool" and click on update.
If that isn't working then you must contact TT because you'll be in a set of circumstances their snazzy new website doesn't allow for... so they might have to manually tweak it for you.