According to several other cuisine websites, the Rinktum Ditty / Diddy / Tiddy is of Welsh origin. If you read the recipe, it looks like a fancied-up Welsh Rarebit / Welsh Rabbit, so that does sound reasonable. Does that argue a Welsh origin for the word itself?

http://music4kidsandmore.com/duckinmillpond.html will give you not only the words but the midi for "Duck in the Millpond", with the chorus of "Lord, Lord, gonna get on a rinktum" (4X)    (Let us hope the 'rectum' interpretation is by now far behind us...)

And apparently the Alka-Seltzer Song Book - "Circa 1937. Condition: Good Plus Wonderful booklet full of period ads and beloved songs of the era...." includes a melody called 'Hi Rinktum Inktum' (with photo of Lulu Belle and Scotty).   Who Lulu Belle and Scotty were, I shall never know - candidates for the plugged product, perhaps, after a steady diet of 'beloved songs of the era'...