Hi
Apple Watch Series 3 (GPS + Cellular) 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.
It is not possible to have service for your Apple Watch from a different carrier to that for your iPhone.
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.
You may wish to contact your carrier to enquire about their plans for providing service plans for iPhone that support Apple Watch Series 3 (GPS + Cellular).
You may also wish to monitor this page for further carriers being added over time:
Apple Watch - Carriers - Apple