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The obsolete exclamative phrase not a word of the pudding was used as a warning or exhortation to say nothing about a particular matter.

This phrase occurred, for example, as follows in A New Dictionary of the Terms Ancient and Modern of the Canting Crew. In its several Tribes, of Gypsies, Beggers, Thieves, Cheats, &c. With an Addition of some Proverbs, Phrases, Figurative Speeches, &c. Useful for all sorts of People, (especially Foreigners) to secure their Money and preserve their Lives; besides very Diverting and Entertaining, being wholly New (London: Printed for W. Hawes, P. Gilbourne and W. Davies, [1699]), by “B. E. Gent.” [s.v. MU, page unnumbered, column 2]:

Mum-for-that, not a Word of the Pudding.

The phrase not a word of the pudding is of unknown origin. The explanation given in The Birth, Life and Death of John Frank (cf. quotation 5 below) is merely a humorous tale. It seems to me that, in this phrase, the noun pudding refers to a sausage-like mass of seasoned minced meat, oatmeal, etc., stuffed into a prepared skin and boiled. Therefore, the phrase may allude to the mysterious nature of pudding stuffing. Likewise, the expression bag of mystery denotes a sausage, with the implication that the ingredients are of a dubious nature.

These are, in chronological order, the earliest occurrences of the phrase not a word of the pudding that I have found:

1-: From Piazza Universale di Proverbi Italiani: Or, A Common Place of Italian Proverbs and Proverbial Phrases. Digested in Alphabetical Order by way of Dictionary. Interpreted, and occasionally Illustrated with Notes. Together with a Supplement of Italian Dialogues (London: Printed by F. and T. W. for the Author, 1666), by Giovanni Torriano [Notes relating to the Proverbs of the letter Z, page 325, column 2]:
—Note: Here, whist is an exclamation to command silence:

10 As much as to say, Whist, mum to that, no more of that, not a word of the pudding, if you love me; Here are Fathers who will over-hear us, and Jobe us.

2-: From Dictionarium Latino-Barbarum. Cui præmittitur præfatiuncula, docens quo pacto Barbaries in Latinitatem irrepserit (Londini: Typis J. C. Impensis Johannis Wright, & Richardi Chiswel, 1677), by the English cleric and lexicographer Adam Littleton (1627-1694) [Linguæ Latinæ Dictionarium Classicum, s.v. muttum, page unnumbered, column 1]:
—Notes: The Latin noun muttum designates a mutter, a grunt; the Latin phrase “proverbialiter dicimus, muttum nullum emiseris” translates as “we say proverbially, you shall not utter a single sound”; ‘Cornut.’ refers to the philosopher and grammarian Lucius Annaeus Cornutus (1^st century AD):

Proverbialiter dicimus, Muttum nullum emiseris, Cornut. Not a word of the pudding.

3-: From The Virtuous Wife; Or, Good Luck at last. A Comedy. As it is Acted at the Dukes Theater, by His Royal Highness His Servants (London: Printed by T. N. for R. Bentley and M. Magnes, 1680), by the English playwright Thomas D’Urfey (c. 1653-1723) [act 1, page 12]:

Beauf. How? he married, and to a young Woman? I must know more of this; Sir Lubberly, a word with you. Prethee what young Lady is it your Unkle has married—hah—
Sir L. Widg. Ha—what is she? why wou’d you know now, wou’d you know—gad it won’t do—ha ha ha! not a word of the Pudding not I—ha ha!

4-: From The English Rogue: Continued in the Life of Meriton Latroon, and other Extravagants. Comprehending the most Eminent Cheats of both Sexes (London: Printed for Francis Kirkman, and are to be Sold by William Rands, 1680), by the Irish author Richard Head (c. 1637-1686) [part 4, chapter 22, page 297]:

Being master of a considerable sum of Money, and being [missing word?] of his London frollicks, he resolved for the Countrey; and providing himself with choice of Geldings, and variety of Rich Clothes for himself, and new Liveries for his two Servants, he took his journey. I cannot give you any particular account of his transactions in the Countrey, because it was at too great a distance; but in general, I heard he plaid over his old freaks, the second part to the same tune: but this I know, that passing for a man of a great Estate, and being plentifully furnished with Moneys, he was admitted into the Family of a Person of Quality; where, however he carryed himself abroad, yet at home he was so civil, as that courting the Daughter, he obtained her and her Friends consent to a Marriage. They questioned not his Estate, (which they knew had been considerable) because he demanded no portion: and thus was our wild Gallant become a staid man, if Marriage would make him so; but although it might operate somewhat at first, yet that good humor held him not long for he was soon weary of any thing that was good; and, as I heard his wife being so, he was the sooner weary of her: and therefore, and because the Money he had brought with him was spent and gone, he was then again for London. I suppose he engaged himself to return speedily; but he who never kept any ingagement, was sure to break that: And now being come to London, he visited all his old Friends; but there was not one word of the pudding; he would not own the alteration of his condition, but had a mind still to pass for a Batchelor; for under that notion he might practise his debaucheries the more freely; for he did intend to cheat any maid, that would be so easie, of the most precious Jewel they had.

5-: From The Birth, Life and Death of John Frank, And of the Pranks and Jests he play’d, though a meer Natural Fool ([London]: Printed by J. M. for J. Deacon, [1682?]) [pages unnumbered]:

How Iack deceived Mr. Sorrel, a Rich Yeoman of Pudings and Links, which hung up in the Chimney.

Jack was often upon the ramble, and one day he went to this Yeomans House, who loved much to make sport with him: the Servants being all busie, some in the Barn, and some abroad among the Cattel, there was none but he and the Fool all alone. Mr. Sorrel says Jack, shall us play at Blindmans-buff; ay saith he, with all my heart, Jack, you shall be blinded, says Jack, that I will, Jack, saith he; so pinning a Napkin about his Head and Eyes, now turn about says Jack; but you see Mr. Sorrel, you see no good faith Jack, saith he, I do not see: Jack shuffled up and down the Ritchin also for to catch him, and still cry’d, you see Mr. Sorrel, you see; but when the Fool perceived indeed that he did not see, he went to the Chimney, and whipt down some of the Puddings into his Pockets, and so he did full as he came to that end of the room, till he had filled both his Pockets and Breeches with them, which was soon done, though they were large, for he was very quick. The doors being open, Jack runs away so fast as he could, leaving the good man blinded, who wondring that he did not hear the Fool, cryed out, Jack, Jack, but finding no answer, he pull’d off the Napkin, and saw the Fool was gone, & that he had taken many of his Links and Puddings with him, he was so enraged, that he sent his Blood-Hound after him, which when Iack did perceive that the Dog came after him, he took a Pudding and flung to the Dog: the Dog smelling to the Pudding, Iack gained ground the while and still as the Dog followed him he threw a pudding back, and so he did till he had got into a House.
This was soon spread abroad to the great shame and vexation of the honest man, that he could not endure to hear of it.
It fortun’d long time after that Mr. Sorrel among other Tenants did go to pay his duty and respects to the Lord where Iack then lived: Iack espying him, went up and told his Lady, that Mr. Sorrel was come: the Lady being afraid that the Fool should offend him by speaking of the Puddings, threatning him saying, Sirrah, if you speak one word of the Pudding, you shall be Whipt: No says Iack, I will not.
But when the Lord and Lady was at Dinner; and Mr. Sorrel and the rest with them, Iack goes to Mr. Sorrel, & shakes him by the hand, saying, how do you do Mr. Sorrel, how do you? then whispering as it were to him, but so loud that all the company did hear him; Mr. Sorrel, says Jack, not a word of the Pudding, Mr. Sorrel; at which they all could not forbear laughing, but the good man was so much ashamed, that the company were sorry to see, and never came there no more. Ever since it is a by word to say (not a word of the Pudding.)

6-: From The Observator (London, England) of Wednesday 5^th April 1682 [number 119, page 2, column 1]:
—Note: This newspaper was written by the English journalist and pamphleteer Roger L’Estrange (1616-1704):

There pass’d an Order for the Removing of the Quakers into Bridewel […]. Finding themselves in the hands of the Law, they are now content to Enter into Recognizance, as Others do; but still more are taken up; and the Reformation, upon the matter of Conventicles, goes Chearfully forward: But this I must Acknowledge, that Mr Weeks’s coming off, Encourag’d the Independents to Meet again, and upon Sunday last was Sennight, they were snapp’d again; and I could tell ye who was taken there; but not a word of the Pudding.

7-: From The History of the most Renowned Don Quixote of Mancha: And his Trusty Squire Sancho Pancha (London: Printed by Tho. Hodgkin, and are to be sold by John Newton, 1687), by the Spanish novelist and playwright Miguel de Cervantes Saavedra (1547-1616), translated by the English author John Phillips (1631-1706?) [part 2, chapter 1, page 450]:

The Dutchess first beginning, now we are alone, said she, I would desire yee, Mr. Governour, to resolve me certain Doubts, which I find very difficult to be understood, in the History of the Famous Don Quixote de la Mancha.
[…]
Sancho having heard the Questions propounded, rose from his Seat without speaking a word, lay’d his Finger upon his Lips, and creeping softly round the Room step by step, like a Nurse in a Sick Man’s Chamber, he took up the Hangings every where, to see that no body listen’d; and then returning to his Seat, Madam, said he, ’tis good to be sure, the Walls have Ears, but now knowing my self to be as safe as a Thief in a Mill, you shall have my very Heart out of my Belly; I am ready to answer all your Questions, tho as numberless as the Sand of an Hour-Glass, without fear, scruple or hesitation; But not a word o’ the Pudding, Madam.

8-: From The Great French Dictionary. In Two Parts. The First, French and English. The Second, English and French; According to the Ancient and Modern Orthography. [&c.] (London: Printed by J. Redmayne, for Tho. Bassett, 1688), by the Swiss lexicographer Guy Miege (1644-1718?) [part 2, s.v. word, page unnumbered, column 3]:

Say not a VVord of that, † not a word of the Pudding, n’en dites mot.

—Note: Guy Miege explained the signification of the dagger symbol (i.e., †) as follows in the Preface to his dictionary:

† Shews, that the Word is only used in a burlesk, jocose, or comical Sense; or else, that it is not current in any Style, but is either forced, or Provincial, or such as grows out of date.