how is it that the narrator finds himself swimming and bathing in the cold water ?

Do You mean George? George thought he was late getting into the city so he bathed in cold water.

And he flung the watch down, and sprang out of bed, and had a cold bath, and washed himself, and dressed himself, and shaved himself in cold water because there was not time to wait for the hot, and then rushed and had another look at the watch.