The important thing to understand is that in the 1920s/1930s there was no 'tuition fee' (as we understand them now) to attend Oxford - rather your outgoings would have comprised living costs, and expenses related to your stay at one of the constituent colleges (known as battels*). These expenses put education at this university out of reach for the vast majority of the population, save the independently wealthy.

I imagine if 'Gatsby' applied to a college that deemed him a sufficient fit, with the ability to demonstrate financial stability in order to stay the course, AND he passed the university-wide mandatory Latin exam before matriculation, then they would allow him entry.

I've not heard of US troops being allowed 'scholarships' at Oxford colleges on the basis of their service, until the latter half of the last century the central University wouldn't have had enough clout to demand colleges accept these kind of scholars. Undergraduate admissions was (and still is) managed by the various autonomous colleges and halls that make up the 'collegiate university'.

Source: Robert Anderson, British Universities Past and Present (London, Hambledon Continuum: 2006)

Recommended reading if you are interested in social developments at Oxford during the long 19th century: Goldman, Laurence (1995) Dons and Workers. Oxford: Oxford UP

*EDIT: Useful blog post from Oxford University Press which explains what battels are, whilst shedding light on some other Oxford lingo.

Just an add-on question, how hard was it (during the time of the Indian struggle for Independence) to get into colleges like Oxford, Cambridge and UCL? Was it as hard as it is today or comparatively easier? (Gandhi attended UCL, if I'm not mistaken.)

Maybe the exact quote from the book would help someone with more knowledge of history than me come to a clear answer.

“It was an opportunity they gave to some of the officers after the Armistice,” he continued. “We could go to any of the universities in England or France.”

I cant speak to whether or not this actually happened in real life but be aware that Gatsby only says that because he needs to find an excuse how he could have went there.

Remember, most of Gatsby's background is a fabrication. Earlier in the story he claimed he went to Oxford because "all his ancestors have been educated there for many years. It is a family tradition." This is when he is still claiming to have come from a wealthy family in the Midwest.

Later in the novel, when pressured about being an Oxford man by Tom Cavanaugh he has to admit he was only there 5 months and now changes his story that he got this opportunity because of his military service. Neither Jordan or Tom are ever convinced he went there. Even Nick, after hearing Gatsby say he "was brought up in America but educated at Oxford," notices Gatsby is uncomfortable about saying as much and believes him to be lying.

TL:DR I am not sure of the validity of this practice, but it is almost certainly a lie by Gatsby to cover his tracks when people realize he couldn't have gone there

Just to make sure, I believe the book says American officers were allowed to do this, not all American soldiers.

A slightly different thing was Bull College, Cambridge. It was founded for American servicemen and women after WW2 and lasted about a year.