There are two commonly used terms "mushroom" and "toadstool". For most people "mushrooms" are the white things in the supermarket, and toadstools are evil and poison that can probably kill you if you touch.
(this is exaggeration, but many British people have no experience of collecting any food from the wild, and seem to think all fungi, except the mushrooms in the supermarket, are toxic).
The word fungus (plural fungi) is also used, with a wider sense, including moulds.
There are many common names for wild fungi. Naturalists and gourmets will know these, you can see them in any guide book. This is no different to Poland, except mushroom collecting is less common in the UK than in the Poland, and so the words for specific fungi are less common.
And just as in Poland, you would say "On this foray, we're going to pick only Cep, and Chanterelle. Don't touch the Fly Agaric under the birch trees, and certainly don't pick anything with a green-brown cap, white gills and a sacklike vulva.