How old is the song? Circa 19th century and earlier, Willy is a common shortening of William and wouldn't have been snigger worthy. Lack of personal alarm clocks also made 'cock crow' a rough means on telling time in the mornings (anyone who keeps poultry may well dispute this.....   :) ). I think the sniggering's the lack of imagination some suffer from......

As to the whole 'folk police' thing, well, it's up to you. Perhaps you can judge audiences as you go along......?