you slip your stocking feet into wellies ~ of troy
Yep! In Britland (or my bit of it anyway), wellington boots are worn straight on the feet just like any other boot, good for dog walking etc. The type of strong rubber boots worn on construction sites are known as gum boots. Overshoes can be called galoshes (seldom heard now) but more usually just ‘overshoes’ and seem to be mainly used for slipping on when just popping into the garden (or yard as USns call it). A quote from Word Detective:
"Galosh" comes from the French word "galoche," which is probably related to the medieval Latin "galopedium," meaning "wooden shoe." One authority believes that the ultimate source was the Latin "gallica," a short form of "Gallica solea," or "Gallic sandal."
WD also claims that Brits use the term ‘rubbers’ for this footware. I have never heard them called that. Rubbers are for deleting stuff, except in the US where they are for preventing stuff.