You’d think English would have a word, but I can’t think of one. Maybe there are dialect words around.
Nearest I can think of is “glory hole”, but I don’t think that has to be under the stairs.
The way our house is designed, there isn’t a c-u-t-s as such. What would be the cwtch opens out in the kitchen, and functions (not terribly well to be honest) as a pantry. The Welsh for pantry appears to be either bwtri (presumably related to English buttery), or the (fairly boring) pantri.
(Although both words have been in Welsh for a good long time, according to GPC).