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The obsolete British-English expression dog in a blanket, also dog in the blanket, designated a rolled jam pudding or currant dumpling.

However, the expression dog in a blanket also designated “a roly-poly suet pudding made with slices of bacon” according to a transcript of the lecture that one Mr. W. H. Hills gave at the Albion Hall, Horsham, on Monday 21^st January 1924, on “Old Sussex Customs, Superstitions and Sayings”—transcript published in the West Sussex County Times, Horsham Advertiser & Sussex Standard (Horsham, Sussex, England) of Saturday 26^th January 1924 [page 6, column 4]:

SUSSEX PUDDINGS.

A “pudding cake” is a composition of flour and water with a little suet boiled, and is flat; if it is round it is a “hard dick.” It is also known as a “swimmer” from the fact that it is generally boiled or steamed on the top of anything else which may be cooking in the same saucepan. Who has not heard of a roly-poly pudding? Many outside our county possibly. It is a pudding made of a paste of flour, fat and water, rolled out flat, jam spread thereon, rolled up, tied in a cloth and boiled. A suet dumpling boiled in a stew or soup is generally termed a “floater,” while a roly-poly suet pudding made with slices of bacon is known as “dog in a blanket.” A real Sussex pudding is composed of a compound of flour and water, with generally a little suet added, made oblong and boiled in a cloth. It is very often eaten when it is hard and cold, with cheese or butter, as bread would be.

These are, in chronological order, the earliest occurrences of the expression dog in a blanket, also dog in the blanket, that I have found:

1-: From The Serious Affliction of Having a Good Appetite, published in The Mirror of Literature, Amusement, and Instruction (London, England) of Saturday 16^th June 1827 [page 402, column 1]:

I was obliged to have recourse to an old woman who supplied the school with apples and nuts, and in whose debt I had fallen “full fathom five” deep, to turn the contents of my library into edible articles. If Cicero could have arisen from his grave, what a shock it would have given to his feelings to have discovered himself transmigrated into a pigeon-pie! How the elegant Horace would have shrugged up his classic shoulders when he discovered a pound of sausages considered better company than himself! and the tender Tibullus would have broken his heart to a dead certainty, could he have beheld his place in my “locker” occupied by a rouleau of “dog in the blanket!”

2-: From An English Dinner Party. As described by a Parisian, published in The Court Gazette, and Fashionable Guide (London, England) of Saturday 2^nd February 1839 [page 691, columns 2 & 3]:

The puddings, more especially, are a favourite entremet [sic]. The genus is rich in species. I took notice of the name of that of yesterday. It is called “a dog in a blanket.” It is a kind of paste, boiled in water, and rolled up with layers of preserves—you would be sure to like it. […]
[…]
[…] As it struck eleven, who should appear but the servant, laden with a tray thrice as large as ever—in short, with a second dinner complete, and ready served up: there was the roast beef, with the preserves, the pie, and the “dog in a blanket,” and the confounded decanters again, brim full to the throat, just as though they had not been emptied over and over again a dozen times.

3-: From Scotch Courtiers, and the Court (Edinburgh: William Whyte and Company, 1842), by the Scottish author Catherine Sinclair (1800-1864) [chapter 11, page 88]:

“The French cook at Taymouth I’m told has elop’d,
“Another from Paris will come it is hop’d.
“Just fancy a Scotch cook brought here from Auld Reeky!
“Boil’d bagpipes and haggis, sheep’s head, cocky leeky.
“A dog in a blanket!—a toad in a hole! *
“I’d rather eat frogs! and indeed on the whole
“Chinese mandarins with their soup of bird’s nests
“Less odious must be, I should think, to their guests!”
* There is an old proverb against giving things a bad name, and certainly these are two oddly named Scotch dishes, the first a fruit-pudding, and the second mutton chops buried in batter, but an English lady was in the last extremity of consternation once, when her Scotch cook proposed them for dinner, which is one instance among thousands what false impressions may be taken up by strangers in an unknown country.

4-: From The Greatest Plague of Life: Or The Adventures of a Lady in Search of a Good Servant. By one who has been “almost worried to death.” Edited by the brothers Mayhew. Illustrated by George Cruikshank (London: David Bogue, [1847]), by the British journalists and authors Augustus Mayhew (1826-1875) and Henry Mayhew (1812-1887) [chapter 12, pages 178 & 179]:

Mr. Edward had been saying, in his nasty mean way, as he never had a pudding or a pie for dinner, he supposed ribbon had got so dear the housekeeping couldn’t afford pastry; so I thought I would put a stop to his shabby satire, and let him have a nice “dog in a blanket,” as a treat for dinner one day—especially as he’s very partial to it; and, certainly, if it’s made with a nice thin crust, and plenty of good strawberry—or even I don’t mind if it’s raspberry—jam, I do think it is as nice a dish as can well be put upon table—only the worst of it is, one’s apt to eat too much of it; and, I don’t know whether my fair readers find it so with them or not, but to me it’s rather indigestible, or, I must say, I should let dear Edward have it oftener.
Accordingly, as, of course, I fancied that silly Emma of mine, blockhead as she is, couldn’t well go making any mistake with so simple a dish as a “roley-poley pudding,” and I didn’t feel much in the humour to go messing with flour in that hot kitchen, I had the girl up, and to guard against mistakes, I asked her whether she knew what a dog in a blanket was?