Phone transfer

I moved from Stirling to Edinburgh five and a half years ago. The phone bills for the old flat are still in my name, as at the time it would have meant about a week of no internet for the people in the flat.

I'd rather it wasn't in my name, but I'm not willing to have them cut off for a week to get it sorted. Anyone had any experience in transferring a phone connection/ADSL from one name to another in recent years?