They did have central heating in the twenties.  It consisted of a furnace in the basement located below a three foot grill in the first floor of the house and connected to the furnace top by a pipe of the same dimensions.

The expectation was that the heat would then rise throughout the house and rise to the second and third floors if such existed.  Your icy cold bathroom is reminiscent of one I remember from the late twenties and early thirties.

Hope these data are useful.

Love  GD