What is the origin of being "in the pudding club"?

Partridge's A Dictionary of Slang and Unconventional English gives this entry:

pudding club, put in the. To render pregnant: low: late C. 19-20. James Curtis, The Gilt Kid, 1936. See also pudding, with a bellyful of marrow.

pudding, with a bellyful of marrow- : in the pudding club. Pregnant: low : C. 19-20; ob. Cf. pudding, n., 2. The latter, esp. as put in the pudden club, to render pregnant, is still current : witness James Curtis, The Gilt Kid, 1936.

That entry (interior emphasis my own) leads us to pudding, entry 2:

pudding 2 n. Coïtion; the penis; the seminal fluid: low coll.: from Restoration days. Wit and Mirth, 1682

So it comes from pudding in the sexual, coital sense given above.

I can't tell you whether it's used today, but it was still current in 1976. I'm watching a rerun of a British sitcom (originally aired in May 1976) in which the phrase was used; the character who used it, and the two who understood it, are urban (London), are lower-middle class economically, and are middle-class in outlook (they value outward respectability and the upholding of middle-class social norms).

In the 1828 AMERICAN DICTIONARY OF THE ENGLISH LANGUAGE you can read definitions for pudding: "what bulges out, a paunch." So a pregnant woman had a "bulge" for her tummy same as a "paunch."

I was just watching "Last Tango in Halifax," a British sitcom, and the phrase "he put her in the pudding club" was used as it if was commonly understood.

It wasn't merely a sexual reference; more of a play on words since there used to be (and still are in a few places) Christmas clubs, in which everyone paid a small amount each week and received a hamper at Christmas, and bottle clubs, in which the reward for your subscription was a bottle of whisky. So presumably the lady in question had received a package as a result of her efforts...

And the shortened version, in the club, is still used in Britain (or in London, at least). I would advise caution, since it is very colloquial bordering on the vulgar; but I would think it would be understood everywhere.