The colloquial British-English noun oojah denotes:
– a thing whose name the speaker cannot remember, does not know, or does not wish to mention;
– and, by extension, a useful implement, a gadget.

The origin of this noun is unknown; there are, at least, three hypotheses:

First hypothesis: The noun oojah is merely a nonsense word; this is supported by the fact that, in the caption to the cartoon published in Punch, or the London Charivari of 24^th January 1917 (cf., below, quotation 1), the noun oojah is associated with tiddley-om-pom and umpty-poo.

Second hypothesis: The noun oojah is a formation similar to nouns such as what-d’ye-call-’em and thingummy (with oo- in oojah representing a colloquial pronunciation of the pronoun who); this is supported by the following facts:
● In Observations of an Orderly (London, July 1917), by Ward Muir (cf., below, quotation 3), the noun oojah is associated with thingummy and what-d’ye-call-it.
● In Gadjets and Oojahs, published in the Nottingham Journal and Express of 18^th September 1918 (cf., below, quotation 4), the noun oojah is associated with thing-um-bob and thingummyjig.

Third hypothesis: “Oojah may come from the East”, wrote the anonymous author of New Slang Crop in British Army, published in the Evansville Journal-News of 20^th July 1917 (cf., below, quotation 2); in fact, one of the possible origins of oojah listed by the Oxford English Dictionary (OED – online edition, March 2022) is:

Urdu and Persian ḥujjat, argument, pretence, excuse, and its etymon Arabic ḥujja, argument, pretext.

The OED forms the hypothesis that the noun oojah is a shortened form of the synonymous noun oojah capivvy, which is perhaps from Urdu and Indo-Persian ḥujjat kāfī fīhi, literally the argument is sufficient, there’s no more to be said about it, composed of ḥujjat, argument, pretence, excuse (from Arabic ḥujja, argument, pretext), kāfī, sufficient (from Arabic kāfī, sufficient), and fīhi, in it, about it.
The following, from Soldier and Sailor Words and Phrases (London: George Routledge and Sons, Ltd., 1925), by Edward Fraser and John Gibbons, seems to support the hypothesis that the noun oojah is a shortened form of the noun oojah capivvy:

OOJAH (also OOJA-KA-PIVI): A substitute expression for anything the name of which a speaker cannot momentarily think of, e.g., “Pass me that h—m, h—m, oojah-ka-pivi, will you?”

The earliest occurrences of the noun oojah that I have found are as follows, in chronological order:

1-: From the caption to the following cartoon by the British painter, illustrator, cartoonist and author Leonard Robert Brightwell (1889-1962), published in Punch, or the London Charivari (London, England) of Wednesday 24^th January 1917:
—This cartoon depicts a N.C.O. (i.e., a non-commissioned officer) talking to a private who is peeling potatoes:

N.C.O. “Here! just grab the oojah an’ dash round to the tiddley-om-pom for some umpty-poo!”
Private (ex-professor of languages) learns later that he was expected to fetch a bucket of coke from the stores.

2-: From the Evansville Journal-News (Evansville, Indiana, USA) of Friday 20^th July 1917—the anonymous author of this article is likely to be Ward Muir (cf., below, quotation 3):

NEW SLANG CROP IN BRITISH ARMY

London, July 20.—An entirely new crop of slang has come into force in the British army during the past year. They have taken the place of “blighty” and the rest of the picturesque synonyms that were uppermost a year or so ago. A hospital orderly writes about them as follows:
“[…] At the hospitals, either in the wards or the recreation room, the collector of outre neologisms would have a happy hunting ground. ‘Pass the oojah,’ says the one-armed man who is playing billiards. What is the oojah? The oojah is any object in heaven or earth; it is the thing which has no name or the name of which you have momentarily forgotten. The one-armed man, about to make his stroke, requires the little twisted wire bridge, mounted on a lead pedestal, that forms the cue’s rest which—poor chap!—he ought to have formed with his lost hand. So he demands the oojah, which is army for what-d’ye-call-it. And his opponent, whose face is so disfigured that he has had to be given a moulded mask to cover part of it, dubs his mask his oojah.
“Oojah may come from the East, with ‘cushy,’ and ‘blighty,’ and ‘bondook’ (a rifle), and ‘Sieda’ (good morning), and ‘burgoo’ (porridge), and a host of other jolly synonyms.”

3-: From Observations of an Orderly: Some Glimpses of Life and Work in an English War Hospital (London: Simpkin, Marshall, Hamilton, Kent & Co., Ltd., July 1917), by the British author and photographer Ward Muir (1878-1927), who, during the First World War, served as an orderly in the Royal Army Medical Corps (R.A.M.C.) in a London war hospital:

A few other slang words which I have come across in the hospital, and which seem to me to bear the mark of the old army as distinct from the new, are: “bondook,” a rifle; “sound scoff” (to the bugler, to sound Rations); “scran,” victuals, rations; “weighing out,” paying out; “chucking a dummy,” being absent; “get the wind up,” be afraid (and “put the wind up,” make afraid); “the home farm,” the married quarters; “chips,” the pioneer sergeant (carpenter); “tank,” wet canteen; “tank-wallah,” a drinker; “tanked,” drunk; “A.T.A. wallah,” a teetotaller (from the Army Temperance Association); “on the cot” or “on the tack,” being teetotal; “jammy,” lucky (and “jam,” any sort of good fortune); “win,” to steal; “burgoo,” porridge; “eye-wash,” making things outwardly presentable; “gone west,” died (also applied to things broken, e.g. a broken pipe has “gone west”); “oojah,” anything (similar to thingummy or what-d’ye-call-it); “push,” “pusher,” or “square push,” a girl (hence “square-push tunic,” the “swagger” tunic for walking-out occasions). The words for drunkenness are innumerable—“jingled,” “oiled,” “tanked to the wide,” “well sprung,” “up the pole,” “blotto,” etc.; but I smell the modern in some of these; their flavour is of London taverns rather than of the dusty barrack squares of India, Egypt, Malta, and Gibraltar.

4-: From the Nottingham Journal and Express (Nottingham, Nottinghamshire, England) of Wednesday 18^th September 1918:

GADJETS AND OOJAHS.
SOME ARMY TERMS, AND THEIR MEANINGS.

Four years ago had you heard either of the above words? Now, if you are in the Army, or if you have friends in the Army, you can hardly get along without at least the first, any more than you can get on without small change in your pocket.
A gadjet is a comprehensive term. There are a variety of gadjets connected with a motor-car; an aeroplane is replete with them—thermometer, barometer, altimeter, and the pitot only knows what besides.
Seasoned campaigners introduce gadjets into their uniform—such conveniences as a watch-pocket under the Sam Brown belt, or “clicker” fastenings to keep down the corners of the breast pockets. One may even hear some technical fad of a senior officer described as a gadjet—though this is an unusual use of the word; and I know of one adjutant of an artillery brigade who was generally referred to as The Gadjet. So you see it means lots of things. A definition is hard to find.

A Tune on the “Oojah.”

If you are looking at some piece of machinery and are puzzled by some detail not part of the main structure, you might correctly inquire, “What’s this gadjet for?” I have heard an air pilot say, “I must go and put him up to the gadjets,” when introducing a brother flier to his type of machine.
An oojah is perhaps a less valuable word, though a very convenient one for tired, lazy, or ignorant people. Its meaning is not very far removed from that of the old-fashioned “thing-um-bob” or “thingummyjig”; but the word is far pleasanter to say and far more euphemistic [?].
As an example of its use—suppose you are sitting in the mess, weary perhaps from a day in an O.P., and you want some music. You know there stands in a corner a mechanical device for producing melody, but for the moment cannot get hold of the name gramophone. So you say to a junior officer, “Algernon, you might give us a tune on the oojah,” and Algernon knows quite well what you mean. You employ the word, in brief, to refer to anything the identity of which cannot be mistaken by your interlocutor.