The colloquial phrase before one can, or could, say Jack Robinson and its variants mean extremely quickly or suddenly.
Here, in all probability the male forename Jack, pet form of John, typifies an ordinary man, and the surname Robinson is a generic personal name.
These are the earliest occurrences of the phrase that I have found, in chronological order:
1-: From Filthy Fashions exposed, by ‘Truepenny’, published in The London Magazine: Or, Gentleman’s Monthly Intelligencer (London: Printed for R. Baldwin) of January 1763—the author denounced “a certain French fashion, which, during the present war, hath gradually crept into this kingdom”, i.e., “the French manner of Frizzlation”:
Monsieur, having, with an inimitable air of gentility, deposited his utensils on the table, and familiarly enquired after her ladyship’s health, begins his operation thus: He dextrously separates from the rest, six hairs near the crown of the head, twists them between his thumb and finger, rolls them up from the points to the root, and, before you can say Jack Robinson, locks them fast in a square inch of paper. He then takes the next six hairs towards the front, papering them up in the same manner, and thus he proceeds in a strait line from the crown of the head towards the nose, till he completes a file (to speak in the military phrase) of ten papers. He then gradually descends towards the right ear, which exactly completes a rank of 30 papers.
2-: From The Conclusion of the celebrated Lecture on Heads, published in The Caledonian Mercury (Edinburgh, Midlothian, Scotland) of Monday 16th September 1765:
And suppose now, neighbour Spriggins, this little drop of milk punch (well come, here’s the king: God bless him) suppose this little drop of milk punch to be the main sea ocean: very well! very well! And suppose these three or four bits of cork to be all our great men of war: very well! […] And then our army all should wear a new uniform; all our horse infantry should wear air jackets; and all our foot cavalry should wear cork waistcoats; and then ye know, why they’d be all over the sea before you could say Jack Robinson!
3-: From a letter in which one Abraham Keel explained how his apprentice, Sam, was press-ganged—letter published in The Leeds Intelligencer (Leeds, Yorkshire, England) of Tuesday 9th October 1770:
Sam and I goes down with a boat of goods to Queenhithe, and we had not been there so long as you could say Jack Robinson, bud down comes a press-gang, and hauls the poor lad out of the boat.
4-: From A Tour through Sicily and Malta. In a Series of Letters to William Beckford, Esq. of Somerly in Suffolk (London: Printed for W. Strahan and T. Cadell, 1773), by the Scottish travel writer Patrick Brydone (1736-1818)—during a dinner in Sicily, a sea captain has been served a painted ice cream in the shape of a peach, which he mistakes for a real fruit until he has “one large half of it in his mouth”:
The violence of the cold soon getting the better of his patience, he began to tumble it about from side to side in this mouth, his eyes rushing out of water, till at last, able to hold no longer, he spit it out upon his plate, exclaiming with a horrid oath, “A painted snowball, by Heaven!” and wiping away his tears with his napkin, he turned in a rage to the Italian servant that had helped him, with a “d—n your maccaroni eyes, you rascal, what did you mean by that?”—The fellow, who did not understand a word of it, could not forbear smiling, which still convinced the captain the more that it was a trick; and he was just going to throw the rest of the snowball in his face, but was prevented by one of the company; when recovering from his passion, and thinking the object unworthy of it, he only added in a softer tone, “Very well, neighbour, I only wish I had you on board ship for half an hour, you should have a dozen before you could say Jack Robison [sic], for all your painted cheeks.”
5-: From Evelina; or, A Young Lady’s Entrance into the World (Dublin: Printed for Messrs. Price, Corcoran, R. Cross, Fiszsimons, W. Whitestone, Chamberlaine, Williams, J. Hoey, W. Colles, E. Cross, Burnet, Walker, C. Jenkin, White, J. Exshaw, J. Beatty, and G. Perrin, 1779), by the English author Frances Burney (1752-1840):
“I’d lay ten pounds to a shilling, I could whisk him so dexterously over into the pool, that he should light plump upon his foretop, and turn round like a tetotum.”
“Done!” cried Lord Merton: “I take your odds!”
“Will you?” returned he; “why then, ’fore George, I’d do it, as soon as say Jack Robinson.”
6-: From a letter published in The Reading Mercury, and Oxford Gazette (Reading, Berkshire, England) of Monday 26th June 1780:
A man who could liberate a diseased leg from matter collected there, and fill it again with matter in a trice, and who could restore a broken Astragalus to a sound state, whilst you could say Jack Robinson, must be a devilish clever fellow!
7-: From A Classical Dictionary of the Vulgar Tongue (London: S. Hooper, 1785), by the English antiquary and lexicographer Francis Grose (1731-1791):
Jack Robinson, before one could say Jack Robinson, a saying to express a very short time, originating from a very volatile gentleman of that appellation, who would call on his neighbours, and be gone before his name could be announced.
8-: From Familiar Letters, from an Elder to a Younger Brother, Serving for his Freedom in the Trinity-House, Newcastle upon Tyne (Newcastle upon Tyne: Printed for the author, by L. Dinsdale, 1785):
Being amongst the broken water, mind ye, a sea took her [i.e., the ship], and made a clean sweep fore and aft,—smack smooth, by Jove!—Away went masts and rigging by the board, and before you coudlst say Jack Robinson washed me, who was holding on by the forecastle bits, over the starboard bow.
9-: From this poem, published in the Sussex Weekly Advertiser: Or, Lewes Journal (Lewes, Sussex, England) of Monday 11th July 1785:
Aminadab to William Pitt.
I Verily unto thee say,
Friend William, commonly call’d Pitt,
Thou’lt find, upon some futnre [sic] day,
That thou art bit.
I ween no safety thou wilt find
In thy uprightness and thy worth;
This Fox is of a subtle kind,
And too far North.
This varlet hath about thee planted
Knaves with whom thou dost advise,
Thy chambers are by devils haunted:
Learn to be wise.
These devils, be they whom they may,
’Twere easy to discover one,
In less time than a man might say
10-: From a letter that Edward Winslow (1746-1815) wrote from Saint John, New Brunswick, to H. M. Gordon on Monday 18th July 1785—as published for the New Brunswick Historical Society in Winslow papers, A.D. 1776—1826 (St. John, New Brunswick: The Sun Printing Company Ltd., 1901):
The transportation of a lusty wife, three little Brats & a large collection of lumber across the Bay of Fundy was no inconsiderable job. That over I was saluted with a very severe fit of the Gout. I found here no preparation for my reception & I was obliged to tumble Mrs. Winslow and the little ones into the crowd that filled the House at Portland-point, & before I could say “Jack Robinson” the ship Parr made her appearance and disembarked my mother and sisters who of course made a considerable addition to the party.