You are talking nonsense, BT can update BT routers , but have absolutely no reason ( or access ) to third party equipment , but third party equipment can ‘call home’ and update themselves if that how the equipment was designed, but that has absolutely nothing to do with BT.
If your service works with the BT router , but doesn’t work with your third party router , it’s petty clear the third party router is mis configured or faulty, if it’s mis configured that’s nothing to do with BT.
As you seem a little unconvinced, head over to your router’s community forum , or a general router forum and ask ‘ Can BT remotely access my non BT router ‘ .
