Hi
If you buy an Apple Watch outright, then you would not also need to pay your phone provider for your Apple Watch.
The built-in cellular features require a carrier plan for your iPhone that specifically includes support for Apple Watch, including the carrier activating the watch on the plan. The watch cannot be set up on a standalone basis or using service plans from different carriers for each of the iPhone and watch.
EE is currently the only UK carrier that supports this feature.
EE's price for adding support for Apple Watch (GPS + Cellular) is the same for all customers. Currently, in addition to the cost of the iPhone's service plan, it is free for 6 months and then 10GB per month for £5 per month:
http://ee.co.uk/help/add-ons-benefits-and-plans/calling-texting-and-data-add-ons /apple-watch-series-3-ee-set-up-and-plan…
If you purchase an Apple Watch from Apple or elsewhere, then you will not also have to pay EE for the Apple Watch itself.
If you do not wish to sign up with EE, then you cannot currently use the built-in cellular feature with another UK network:
Apple Watch - Carriers - Apple
Apple Watch Series 3 (GPS + Cellular) can also be paired, set up and used for other features whilst skipping the steps to activate cellular.
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.