If you buy a non BT router it will not transmit the BTWifi signal. If you do buy a new router and you are still using the white Openreach modem, make sure that the router has a WAN socket.
If you want to do away with the Openreach modem you will need to get a VDSL Modem/router.
There are no cons to using a non BT router except if your router goes faulty you will not get any BT assistance to fix it.