"Faster than you can say Jack Robinson" (meaning very quickly) is a common English expression, from well before C. S. Lewis's time to the present day. The online Oxford English Dictionary (paywalled but you may have access through your public or university library) has citations ranging from 1763 to 2016. Quoting from the Wikipedia page for Jack Robinson (mythical person):

Jack Robinson is a name present in two common figures of speech. When referring to Jack Robinson, it is used to represent quickness. In contrast, the phrase "(A)round Jack Robinson's barn" has the opposite connotation, implying slowness, as it is often used to refer to circumlocution, circumvention, or doing things in roundabout or unnecessarily complicated ways.

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Multiple citations explain references to Jack Robinson as meaning quickness of thought or deed. The normal usage is, "(something is done) faster than you can say Jack Robinson", or otherwise, "before you can say Jack Robinson". The phrase can be traced back to the eighteenth century.

Attested occurrences of the phrase “say Jack Robinson”

A Google Books search reveals a cluster of published writings from the early 1760s that include the phrase “say Jack Robinson.”

From “Another Letter of the Sailor from the Havannah,” in The Edinburgh Magazine (November 1762):

I was sorry for the commodore of the castle, cause he was a brave fellow ; a ball came aboard of him, under the larboard-side of his breast, and clapped a stopper upon his commission, before one could say Jack Robinson.

From "Truepenny," letter about “a filthy fashion,” in The London Magazine (January 1763):

You cannot but have taken notice, sir, you who are so universally conversant with the ladies, that of late, there appears to be an additional growth of hair on the heads (I say, Sir, on the heads) of such of our females as are commonly seen in places of public entertainment : There seems, since the present fashion, to be an additional quantity both in front and rear. Now possibly you imagine this increase to be owing to some newly discovered pomatum, bear’s grease or something of that sort.—No such thing. It is entirely owing to the French manner of Frizzlation. Perhaps you have no idea how this is performed. I’ll tell you, Sir.—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.

From “Lecture on Heads,” in The Gentleman’s Magazine (September 1765):

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 waist-coats; and then ye know, why they’d be all over the sea before you could say Jack Robinson!

From “Tristram Shandy,” Miss C——y’s Cbinet of Curiosities; or The Green-Room Broken Open (1765):

I love vastly an Author that has the Bravery of combating popular Prejudice. There is great Merit in being singular ; and, I trust, such is the Force of our impartial Critic’s Observations and Arguments, that as soon as the Work is published, the World will change their Opinions in an Instant,—in the Twinkling of an Eye—as soon as one can say Jack Robinson—and I hope also, that in Consequence of the humane ZANGA being looked on as the best Actor, the cruel, Browbeating, swearing Roscius will be dethroned.

Theories of the phrase’s origin

The earliest collection of sayings to include “say Jack Robinson” is Francis Grose, A Classical Dictionary of the Vulgar Tongue (1785):

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.

But most lexicographers have been unenthusiastic about this derivation, and some have floated alternatives to Grose’s theory.

From James Halliwell, A Dictionary of Archaic and Provincial Words Obsolete Phrases, Proverbs, and Ancient Customs, from the Fourteenth Century, second edition (1852)

Before one could say Jack Robinson, a saying to express a very short time, said to have originated from a very volatile gentleman of that appellation who would call on his neighbours, and be gone before his name could be announced. The following lines “from an old play” are elsewhere given as the original phrase,—

“A warke it ys as easie to be done

As tys to saye, Jacke ! robys on.”

The source mentioned as having “elsewhere given” the couplet “as the original phrase” appears to be William Carr, The Dialect of Craven, in the West Riding of the County of York, volume 1 (1828):

JACK ROBINSON, What a strange perversion of words will time frequently occasion! “As soon as you can say Jack Robinson,” is a phrase common in every part of the kingdom, but who could suppose that it is a corruption of the following quotation?—

“A warke it ys as easie to be doone,

As 'tys to saye, Jack ! robys on."

Old Play.

Pishy Thompson, The History and Antiquities of Boston (1856) alters the couplet's second line a bit and adds an approximate date for the old play, without saying where that information came from:

“Before you can say Jack Robinson.”

“A warke it ys as easie to be doone

As ‘tis to saye, ‘ ‘Jack’s-robys-on.’”

Old Play, about 1580.

What is this old play? An item in Notes and Queries (1916), after citing Halliwell’s discussion, asks this very question [combined snippets]:

Is the reference [that is, the source of the couplet] known, and what is indicated by “elsewhere”? Does “robys on” mean “Robyson” or “Robinson” or something else?

A query about the saying appeared at 1 S. vi. 415. Beyond an editorial note giving the quote from Grose, I think that there was no reply.

The editor of H.D. Thoreau, Walden: A Fully Annotated Edition (2004) attributes the couplet not to an old play from circa 1580 but to an eighteenth-century song:

From a popular 18th-century song written by a tobacconist named Hudson: “A warke it ys as easie to be done / As tys to saye, Jack! robys on."

Unfortunately, this note appears to garble the song theory and the old play theory, which in several reference works of the nineteenth century appear contiguously and without a clear transition from one to the other. Henry Reddall, Fact, Fancy, and Fable: A New Handbook for Ready Reference on Subjects Commonly Omitted from Cyclopedias (1892), seems to support the song theory:

Jack Robinson. “Before you could say Jack Robinson.” This current phrase is said to be derived from a humorous song by Hudson, a tobacconist in Shoe Lane London. He was a professional song-writer and vocalist, who used to be engaged to sing at supper-rooms and theatrical houses.

But Ebenezer Brewer, The Reader’s Handbook of Famous Names in Fiction, Allusions, References, Proverbs, Plots, Stories, and Poems, revised edition (1910), offers a crucial (and problematic) detail about Hudson:

Jack Robinson. This famous comic song is by Hudson, tobacconist, No. 98, Shoe Lane, London, in the early part of the nineteenth century. The last line is, “And he was of before you could say ‘Jack Robinson.’” The tune to which the words are sung is the Sailors’ Hornpipe.

And finally, as noted in Barrie England’s answer, there is a theory that “say Jack Robinson” can be traced to an allusion in Parliament by Richard Sheridan to another Parliamentarian. However, the John (Jack) Robinson of that anecdote first became an MP in 1764. The phrase thus appears to have existed well before Sheridan had occasion to use it as a sly way of accusing John Robinson, by name, of bribery.

Conclusion

To summarize, the candidates put forward as the true source of a phrase that began appearing in British magazines by 1762 are (1) a song from the early nineteenth century, (2) a Parliamentary riposte from approximately 1780, (3) a mysterious old play first mentioned—but neither identified nor dated—in 1828, and (4) a gentleman identified by name only who was said—in 1785—to have been famous for paying very short visits to people. Theories (1) and (2) fail for simple reasons of chronology. Theory (3) would be far more plausible if anyone had ever been able to identify the “old play” or even to find an earlier reference to it than Carr’s in 1828. And theory (4), though hardly disprovable, offers little in the way of verifiable evidence to support its conjecture.

No wonder Robert Hendrickson, The Dictionary of Eponyms (1972) writes:

Notable attempts have been made to trace this eighteenth-century British phrase, all unsuccessful. … But Jack Robinson was probably used in the phrase simply because it is a very common name in England and is easy to pronounce.

That the first four matches for the word in a Google Books search are from 1762, 1763, and 1765 (two occurrences) suggests that the phrase became popular fairly suddenly, which in turn raises doubt as to how long it went unpublished before that time.

It means quickly. "Before he could say 'Jack Robinson', the meal was cooked". It is an older English expression - I haven't heard anyone say it for years.

Hahaha yes it is outdated.