From Bridewell (named after a well dedicated to St Bride), an area of London that once had a "house of correction".

bridewell (plural bridewells)

1. (sometimes capitalised, dated in British, Ireland, rare elsewhere) A small prison, or a police station that has cells.
   • 2002, Joseph O'Conner, Star of the Sea, Vintage, published 2003, page xix:

     Two Fermanagh girls who never laughed were certain that he must have served time in a bridewell, so cold was his expression and so calloused his small hands.