BT own the socket in the wall, and the device that is your broadband “hub”. That’s all.
On the back of your hub there should be 4 sockets - mine are yellow. These are copper “cat5” network sockets. Network cables are thin, flexible, and can usually be hidden easily. If you can run a cable from your hub to some other location in the house, you can put an access point there. For example, if the hub is in the kitchen, you might be able to get the cable through to the other side of a solid wall and site the point there, which will make a huge difference to reception.
Pikey IT approach - give the new AP the same ID and password as the hub, so devices can migrate automatically. It will work, time critical stuff won’t work - e.g. wandering around the house while making a Skype call.
Proper approach - get 2 APs with hand over features, site one near the hub, the other as far away as possible, have them both set up with the same SSID and password, and disable the BT WiFi. This will be perfect - when you switch AP, there will be almost zero drop out.
With APs, you get what you pay for. Consumer grade stuff is generally pretty awful. All of the problems I have had in the past with connections have been solved by the new APs I’ve put in. Yes, they’re £120 a pop, but worth it. They do PoE as well, so you don’t need to find a mains socket where you want to put them, just the network cable.