Although you are moving into your mothers address , as far as the process is concerned you may as well be moving into a strangers house , instead of a second line for your broadband, which is probably what you should have enquired about , ( but it would be more expensive, you paying for a line and broadband and your mum continuing to pay for her line ) but you are taking over her existing line , and adding broadband to it , if this were a strangers house you wouldn’t want the phone number they had , plus in many cases the existing occupant moving out would want to take the number with them anyway ( if that were possible ) .
Its almost certain that your mum will get a ‘sorry to see you leave’ letter confirming her service is going to stop , and your provider , ( even both are BT ) will provide your broadband service with a different phone number , that’s what happens , the other option would be to add broadband to your mothers line , but then you may be liable for early termination charges at your old address if you were in contract , plus obviously the amount your mum would pay would increase to cover the broadband
