Do they exist in Protestant churches built after the reformation (or is there a continuing tradition of them being made for Catholic churches built since?)

Relatively few churches were built in England from Edward VI's time until the Restoration, by which time the building style had changed from the Perpendicular style- the nice clean update of the "Gothic" or Decorated style which is the one commonly associated with mediaeval churches- to the rather florid "English Baroque" style in which niecties like gargoyles and green men had no place.

Catholic churches were of course not built until the 19th century emancipation, by which time they wouldn't want to be associated with the mediaeval stuff- though they took that up again, as did the CofE, with later Victorian romanticism, like the Oxforn Movement.

So I thgink you can be pretty certain any Green Men are either mediaeval or toy.