It would appear that the story was unfinished and most decidedly not intended for publication. It was, we learn in David Leavitt's The Man Who Knew Too Much: Alan Turing and the Invention of the Computer, written largely as a psychological exercise, likely at the urging of Turing's shrink. You can see the papers here, in the Turing Archive.
Note that the titular character (Alex Pryce) is a thinly veiled version of Turing himself, with his lover cast in the same role. Leavitt offers some commentary to cover the bits of the story that are missing, illegible or glossed over by Turing. The MacGuffin that Pryce is so proud of isn't mentioned in any particular detail but seems to be some kind of satellite.
For ease of reading, I've placed the quotes from Turing's story into italics.
Alec always felt a glow of pride when this phrase was used. The rather obvious double-entendre rather pleased him too. He always liked to parade his homosexuality, and in suitable company Alec would pretend that the word was spelt without the "u." It was quite some time now since he had "had" anyone, in fact it was not since he had met that soldier in Paris last summer. Now that his paper was finished he might justifiably consider that he had earned another gay man, and he knew where he might find one who might be suitable.
Like Turing, Alec has a habit of going on "rather wildly to newspapermen or on the Third Programme." Like Turing, he was also rather untidy, dressing in "an old sportscoat and rather unpressed worsted trousers." One hears the voice of Dr. Greenbaum in the background of the analysis of his sartorial tendencies that comes next:
He didn't care to wear a suit, preferred the "undergraduate's uniform," which suited his mental age, and encouraged him to believe he was still an attractive youth. This arrested development also showed itself in his work. All men who were not regarded as prospective sexual partners were fellow scholastics to whom Alec had to be actively showing off his intellectual powers.
As the story begins, Alec is Christmas shopping. He is also, presumably, looking out for the "gay man" he feels he has "earned" —and at this point the story's point of view changes rather suddenly to that of "Ron Miller," the stand-in for Arnold [Murray]. Ron, we learn,
"had been out of work for two months, and he'd got no cash. He ought to have had 10s or so for that little job he had helped Ernie over. All he had had to do was to hold the night watchman in conversation for a few minutes whilst the others got on with it. But still it wasn't safe." Ron is also "very hungry and rather cold in the December weather." And, apparently, he is not unwilling to consider bartering sex for cash:
If he let someone take him under the arches for ten minutes he might get four bob. The men didn't seem to him so keen for it as they were a year ago before the [Ron's?] accident. Of course it wasn't the same as having a girl, nothing like it, but if the chap wasn't too old it wasn't unpleasant. Ernie had said how his chaps would make love to him just as if he were a girl, and say such things! But these were toffs. Ernie with his pansy [illegible] and his pretty-pretty doll's face could get them as easy as [illegible]. Should think he liked it quite a lot too, the sorry little swine. Heard [illegible] boast he couldn't do anything with a girl when she paid him!
Presumably, then, Ron sees himself as being in a different class from Ernie, that "sorry little swine" - just as Arnold saw himself as being in a different class from such friends as, perhaps, the "little swine" Harry. Nonetheless, he is on the lookout and notices that Alec is watching him:
That chap who was walking round the place had given him quite a look Here he was coming round again. This time Ron stared back, and Alec followed in his walk and hurried on round the plot again. No doubt now of what he wanted. Didn't seem to have the nerve though. Better give him a little encouragement if he came past again. He was coming too. Ron caught Alec's eye and gave him a half-hearted smile. It was enough though. Alec approached the park seat; Ron made room for him and he sat down. Didn't seem to be very well dressed. What an overcoat! Why wasn't he saying anything? Could he be mistaken? No, he was having a furtive look. . .. [I]f he wasn't careful nothing would come of it.
Ron now asks whether Alec has a cigarette. And, as it happens. Alec does—though this requires some explanation:
He didn't smoke, because he hadn't quite enough control if he did, and anyway it didn't really agree with him. But he knew that if he "clicked" he would need some cigarettes. "Doing anything this afternoon?" Alec asked suddenly. It was a standard opening. A bit brusque certainly, but he hadn't thought up anything better: anyway the brusque-ness tended to prevent misunderstandings. This chap would do well enough. Not a real beauty, but had a certain appeal. Beggars couldn't be choosers. He was shaking his head. "Come and have some lunch with me."
Beggars cant be choosers. What is so sad about this moment is the forcefulness with which the need for Forsterian connection obliterates standards, class, memory (of the idealized Morcom), even self-esteem. And not only for Alec—also (though to a lesser extent) for Ron:
"Don't mind if I do," said Ron. He didn't go for lah-di-dah ways of talking. It'd come to the same anyway. Bed's bed whatever way you get into it. Alec thought otherwise, and was silent for a couple of minutes as they made their way towards Grenkoff's [?]. He'd got to go through with the lunch at least now. Ron was quite clear about this too. He was sure of the meal. He wasn't sure whether he'd do anything. Perhaps he'd be able to get something without.
In the restaurant, however, Ron is too dazzled by having "the door opened for you by a commissionaire, and to go through first like a girl," to take much notice of Alec. Instead, his attention is "concentrated entirely on the restaurant and its trappings." And this makes Alec happy: "Usually when he went to a restaurant he felt self-conscious, either for being alone, or for not doing the right thing. Ron wouldn't..."
And here the story breaks off. We never learn what happens to Ron and Alec. They are left forever on the brink of possibility—perhaps of possible happiness—untouched by the shadow that had swooped down and destroyed the life of their creator.