I wonder which came first: the above poem/song, or the catch phrase "does your mother know you're out?"

From Slang and Its Analogues Past and Present, Volume 2 by John S. Farmer and William Ernest Henley ([London:] Printed for subscribers only, 1891), page 300:

DOES YOUR MOTHER KNOW YOU'RE OUT? phr. (streets').—A popular locution, vague as to meaning and inexact in application—an expression expressive of contempt, incredulity, sarcasm, anything you please.—See ALL MY EYE, STREET CRIES, and infra.

ENGLISH VARIANTS.—Has your mother sold her mangle? Not to-day, or it won't do, Mr. Ferguson! Sawdust and treacle! Draw it mild! And the rest! Who are you? All round my hat! Go it, ye cripples! Shoo, fly! How does the old thing work? Well, you know how it is yourself! How's your poor feet? Why, certainly! I'll have your whelk! Not to-day, baker, call to-morrow, and we'll take a crusty one! Do you see any green in my eye? Put that in your pipe and smoke it! Where are you going on Sunday? Go to Putney! Who stole the donkey: the man in the white hat! Cough, Julia! Over the bender! There you go with your eye out! etc., etc.

FRENCH VARIANTS.—Et les mois de nourrice (=and the rest!); du combustible (popular:=go it you cripples); tu t'en ferais péter le cylindre (popular:=don't you wish you may get it); chiche! (popular: a defiant refusal); chaleur! (popular: expressive of contempt, disbelief, and ironical admiration); croyez ça et buvez de l'eau (popular:=believe that and drink water); à Chaillot (=go to Bath and get your head shaved); tu t'en ferais crèver (pop.:=don't you wish you may get it); colle-toi ça dans l'cornet (pop.:=put that in your pipe and smoke it!) je la connais (pop.:=do you see any green?); j'entrave pas dans tes vannes (thieves':=you don't take me in); de la bourrache! (popular:=no go); un sale truc pour la fanfare (popular: an expression of disgust); de quoi (popular: what next? also=wealth, money, etc.); allez donc raconter cela à dache (thieves':=tell that to the marines!); des dattes! (pop.:=take a carrot!); et ta soeur (popular: indicative of refusal, contempt, and insult); faut pas m'la faire (popular: Walker!); et le pouce (pop.:=and the rest!)

1841. Punch, vol. I. p. 6, col. 2. Where are they that should protect thee In this darkling hour of doubt? Love could never thus neglect thee! DOES YOUR MOTHER KNOW YOU'RE OUT?

1864. Sun, 28 Dec. 'Review of Hotten's Slang Dictionary.' Ridiculous street cries, such as DOES YOUR MOTHER KNOW YOU'RE OUT? Or, Has you Aunt sold her mangle? or, You don't lodge here, Mr. Fergusson—whatever those sapient remarks may mean.