In theory you should be able to transfer your broadband from one address to another pretty much seamlessly, and on odd occasions it does actually work like that.
More often though you will find out in practice that, because you are dealing with BT, who have now split themselves down into several smaller companies that appear to have no decent lines of communication between themselves that it will all go wrong. Each department will then blame the other one for the mistakes and it will take days, if not weeks of phone calls to get it sorted out.
Welcome to the joys of dealing with BT, Good luck!