We have put together a short survey to capture your feedback in terms of what BT can do when you experience home Wi-Fi issues. We would appreciate it if you could take a couple of minutes to complete a short survey. Read more from here, Fix my home Wi-Fi
If it is a locked phone you will need to get the unlocking code from your previous provider. If it is unlocked you will need to go into the settings and ensure that the "Mobile Networks" > "Network Operator" is set to BT.
I think your problems are related to the phone having been on the Verizon network.
Have a look at this link and work your way through it to see if there is a setting you have missed.