Check your contract with AOL on how much notice you need to give for cancelling. AOL are utter scum and will try and tie you in for a year. You might need to prepare for some stress!
Once you have a cancellation date, order with Tiscali letting them know when you want it to start. You will need some sort of tag number from AOL also so that Tiscali can pick up your ADSL...
I would expect your broadband to be disconnected for a couple of days. These things rarely go smoothly. AOL don't want to lose you as a customer and will be as obstructive as they can.