OK, so when there is no sync, lift the handset, dial 17070, then option 2 for a quiet line test.
Keep listening to the phone, as the router syncs up do you here 'fax like' tones. If you do, then it is almost certainly an HR (high resistance) fault, there is a bum connection on the pair of wires somewhere. If not then you may be looking at an issue with the DSLAM card in the exchange. My money is on the first option.
Don't be fobbed off by your broadband service provider, with them telling you to ring BT voice faults. Just report it as no sync, jump through their various hoops, and get an SFI booked. You have ruled out that it's anything to do with your equipment, so they can't be passing a charge on to you.