In all these cases there is no link to the animal
Cock - A rooster
The body part is probably from cock to mean raise up, as in "cocking a gun". The bird is old English/Scandanavia
Pussy - A cat
The body part is much older from Scandanavian/OE word for a pocket or bag. The cat is newer and probably just a childish onomatopoeia for soft/furry
Tits - A small bird
The body part is from teats and the French Teton. The bird is from Scandanavian for any small animal, cf Titmouse
Ass - A donkey - okay I see this so this can be taken out.
There is no conection between the words, The body part is OE ærs, Assa is latin . See Why are the "donkey" and the "butt" both named "ass"?
The OED's first quotation for a bombshell describing a woman is 1942, but it was indeed the title of a 1933 film starring Jean Harlow. In fact, the film was later renamed to The Blonde Bombshell so it's possible Harlow got the nickname from the movie.
(It appears in a number of 1933 snippets in Google Books which could be misdated, but look correct.)
The earliest verifiable example I found is in the Spokane Daily Chronicle - Nov 24, 1933:
But it seems that Mr. [Lee] Tracy stepped out on a balcony in his pajamas and made wild gestures during a Mexico City parade, and now Mr. Tracy, a star with a long list of hits from "Blessed Event" to "The Blonde Bombshell" behind him, is no longer with at least not until things are "straightened out," if they ever are.
And from the UK:
R, H., 1933, Dec 09. THREE NEW FILMS. The Manchester Guardian (1901-1959), 16:
Two of them, "The Cradle Song" and "L'Ordonnance," are by famous authors, the first Sierra and the second Maupassant, and have European stars, but the only completely successful one is the one with the least pretension, "Blonde Bombshell," at the Empire.
The next day's Observer explains the name.
LEJUNE, C.A., 1933, Dec 10. The Pictures. The Observer (1901- 2003), 12. ISSN 00297712:
And it is worth mentioning, in passing, that Hollywood cannot be blamed for the title. It is we, the audience, who have spoilt an arresting and altogether unusual title with that ridiculous adjective. When the film first came out in Hollywood, it was called "Bombshell," tout court. But the public stayed away because they thought it was a war picture. "Blonde Bombshell" was the final compromise between dynamics and security. ...
The story is nothing but a snatched handful of experiences form the life of Lola Burns, star of stars in the Monarch studios. She is their "It Girl," or, alternatively, their Blonde Bombshell--eruptive, starry-eyed, warm-hearted, a little fatuous.
Edit: Bill Mullins found slightly earlier references:
The San Diego Evening Tribune 1 Nov 1933 p 8A col 5 has an ad for the
film (titled "Bombshell") with the line "A Blonde Bombshell in the
Picture That Has Thrown Hollywood Into Consternation!"
The LA Times of that day has a small ad (sec II p 8 col 5) that reads:
"Last Times Today Jean Harlow Lee Tracy "The Blonde Bombshell" as if
that were the film's title.
Best Answer
Etymonline lists the origin of shucks thusly:
So, the -s termination comes from the plural in the original, longer expression.