If you are on Tiscali's full LLU package then you would have to get your BT line up and running first before ordering O2. Just contact BT and ask for return to donor or reverse migration, should be free.
Once your line is on BT your current broadband will cease and it may take up to 10 days for BT's database to recognise your number and for O2 to connect. Remeber to confirm with Tiscali you have left them, should be done automatically but best be on the safe side.
If you're not on LLU then usual MAC procedures apply and you could change phone providers at any time.
Contact Tiscali's cancellation line - 0845 077 4488 (may still work) and see what they say.
There's always going to be some overlap in payments unfortunately.