What European power adapter works best with my Xbox 1 S (Battlefield 1 edition)

any power adapter capable of the loads of a game console would likely cost more than the game console

The Xbox One S is multi voltage, so a power cable from Poland that fits into the console should work fine.

They key is to make sure that is has a transformer (that one you linked does as it converts the voltage), and that it will allow for the NA standard plug into it. As well as the plug going out will match up to whatever the European style socket you will
be using. They work for any device not just Xbox's. That one however does not seem to be able to accommodate a European socket so far as I can tell. I think it is an external PSU to replace a dead one in a console that will not power on. In any event I do
not believe that will do what you need it to. Search Amazon for a "travel adapter" there are many different types and brands. You want to ensure it will down volt 220V to 100V and accommodate the NA socket and have a male plug end that will accommodate the
socket type you will be using. (Same link as before, but I made it a proper hyperlink. This one should work for you. However there's plenty of other options that do less and cost less, that could also fit your needs)

https://www.amazon.ca/BESTEK-Universal-Converter-Charging-Worldwide/dp/B01E140XWA/ref=sr_1_23?dchild=1&keywords=Travel+adapter+NA+to+European&qid=1591579232&s=electronics&sr=1-23

There is a
"Storm Grey" Battlefield 1 Edition Xbox One S, but it doesn't seem to be available for sale on the UK Xbox website for some reason.

That's the only Xbox One S version that comes in Black/Grey, as far as I know. Try to shop around for that edition.

What is this adapter called and does it work on xbox 1 S and 1X wired jack controllers or just the original first generation xbox one (not s not x) non wired jack input controllers adapter

thanks

Happy New Year