Hi
Apple Watch Series 3 (GPS + Cellular) is not locked to any specific carrier and has a built-in eSIM that copies information from the SIM card on the paired iPhone when setting up the built-in cellular feature.
This requires a service plan for your iPhone that specifically includes support for the Apple Watch, including the carrier activating the watch on the plan.
Within the UK, EE is currently the only carrier listed by Apple as offering a compatible service plan:
Apple Watch - Carriers - Apple
If you do not wish to sign up with EE for your iPhone at present, then you cannot currently use the built-in cellular feature with another UK network.
However, Apple Watch Series 3 (GPS + Cellular) can still be paired, set up and used for other features whilst skipping the steps to activate cellular, offering the same functions as a GPS-only model.
The cellular feature can then be enabled later, if and when you are using a compatible service plan on your iPhone, whether from EE or another UK carrier in due course.
I suggest that you contact BT Mobile to ask about their plans for offering a compatible service plan.