To understand how you would become an officer you must understand the general clusterfuck that was Britain raising a conscript army in 1914/15.
Britain, unlike many other armies at the time, did not distinguish if regiments as operational units but in an administrative fashion as the parent body of multiple battalions. As a result of this when the war started each regiment had its own strictly defined recruiting areas within Great Britain and Northern Ireland with depots which had its HQ's and training areas and so forth; it was a strictly defined and regimented system. Larger counties had several regiments and small counties had to combine together at times to form a single regiment. Basically as it worked, as Britain was still a professional army at this time, each regiment would be comprised of two battalions of 1000 men. One battalion would be out and about in the world and one would remain in the homeland to refresh/cycle out the men abroad as needed.
Once the Regular Army was sent to France in early August 1914 the only remaining force were the Territorials who had certain legal rights. They could not be sent abroad forcibly but would have to volunteer giving a "General Service Obligation" and they couldn't be transferred from one unit to another, ie, being redeployed into the regular army to work around that. However, with that, it was an almost universal "yes" to go off to war when the call was made. By Christmas 1914 22 Territorial battalions were in France and by February 1915 26 more had joined them.
Kitchner was faced by his first serious issue though -- he had nothing left to build an army with when it became apparent the war wasn't going to be over quick. The entire army was either in France or sailing from the edges of the empire to go to France. He decided upon a new experiment. He would not siphon men from existing formations and basically "raise" armies assisted by men in the current army as was traditionally done prior. Those men were too valuable and already fighting. He would raise an entire army -- NCO's and officers alike -- from scratch. This would be called Kitchner's Army, or, more aptly, "The New Army."
Parliament would pass a bill sanctioning 500,000 men to be raised in 18 new divisions; it had officially begun. Kitchner would give a famous speech calling for "the first hundred thousand", the bravest of them all to step up first and get into the fighting as soon as possible and the nation ate it up. The people were enamored by this idea of a peoples army. Many people genuinely did feel the Empire was at risk and it was their duty and it being in their capacity to personally change the empire's future for the better. While the older men were generally more influenced by pride and patriotism for their 'in danger' empire, the younger felt a need of adventure and travel when most people in Great Britain had not been outside of their town let alone their county. What reason had they not to, on top of that? To quote Middlebrook:
This was an England where a boy left school at thirteen and, if he could get work at all, went into the local factory, mill or pit to work between fifty and sixty hours a week for 5s [shillings], or less. The average wage for a man was under £2 per week and for this he had to work from 6.0 A.M. to 6.0 P.M. with thirty minutes for breakfast and an hour for lunch.
Thousands of boys below 19 (the minimum age) would swarm recruiting depots and be accepted knowingly by recruiting sergeants who collected bonus' for the amount they enlisted. Corporal J. Norton of the 8th Norfolks tells us:
"I was a member of the village cricket and football teams and nearly every one of their members enlisted. I was only sixteen but I tried to join up, too. The recruiting sergeant asked me my age and when I told him he said, "You better go out, come in again, and tell me differnet." I came back, told him I was nineteen and I was in."
Recruiting officers would be opened at police stations and town halls as local authorities and prominent figures would equip and fit the men knowing the War Office would eventually take responsibility for the men they outfitted and reimburse the costs. Dozens of battalions were formed locally as men from counties and even towns would all gather and form their own battalions to serve together. These battalions would be created so rapidly they would name themselves before the War Office could give them official titles. For instance in the industrial areas of the North the men coming from there would be known as "Pals Battalions" and these Pals would all be of a particular occupation. The 3rd Manchester Pals was recruited from Clerks and Warehousemen of Manchester for instance.
Hull would raise four battalions which would serve as the cities own brigade. One of these brigades was comprised entirely of business offices and one was a sportsmen's battalion and another a trademen's. The last, for a lack of a better name, was known as "T'others." These names, comically enough, would stick and even become pseudo-official. All over the nation counties and towns would compete with each other. When in Grimbsy the headmaster of a local grammar school tried to raise a company of 240 men so many men came from surrounding villages 2 battalions would be formed in short order and be known as the Grimsby Chums.
This system encouraged much camaraderie and vigor in the recruitment but it was also inconsistent as you may imagine. For instance, miners from mining towns made fantastic soldiers. They were used to grueling work and dangerous underground conditions which made them perfect for maintaining trenches but in for instance one Yorkshire battalion it had so many miners that they couldn't find any orderly clerks while others had over 900 of the 1000 men qualified to perform the 'paper pusher' jobs but few soldiers.
Lieutenant Colonel R. Fife of the 7th Green Howards recalls:
"Nobody who had anything to do with Kitchner's Army will ever forget . . . when vast hosts of men, without officers, without N.C.O.'s, without uniforms, arms, camp equipment, rations, tents, or anything else except the clothes that they stood in, were assembled in open spaces called camps, and there embodied units of the British Army."
Within three weeks the 500,000 men mark had been met with entirely volunteer conscripts and it was still shooting well past that. Now we get to your question specifically OP: who would lead these men? There were but 500 Indian officers on leave and 2000 "young gentlemen" as they were called to act as junior officers comprised entirely of university graduates. More recent retirees would join up to lead men again in patriotic fervor, however, these would not go very far as you might imagine. When the Pals began to form a battalion was lucky if they could get one Regular Army officer, retired or not, to serve with them. The remainder (a battalion needs 30) were chosen basically at random from businessmen and professionals who had joined.
To really beat how much of a clusterfuck this was let's look at one division, the 21st. Every one of the battalion commanders had been retired when war had been declared and of the other officers none but 14 had any army experience at all (but not with combat). The remainder, over 400 officers total, were newly commissioned and no more special than the conscripted serving as privates. This distinction would be even more acute with Non-Comissioned Officers. All of the reservists brought the NCO's into the Regular Army...now in France. Lieutenant Colonel A.P.B. Irwin of the 8th East Surreys recalls:
"Then the whole battalion was paraded and an appeal was made for anybody who had ever been in charge of anyone else, or wanted to be. About forty men stepped forward; we tied white tape around their arms and made them lance-corporals^[1]."
They literally were taking anyone who was educated or a successful businessperson with any form of leadership experience who said "I want to be an officer". If they for some reason were determined not qualified enough for officer they would at the very least be made NCO. The answer to your question of how would you become a British officer, at least for the raising of Kitchner's Army (before general conscription in 1916), would be "If you wanted to and were relatively intelligent."
It was an imperfect system though. A new battalion, which ideally would have 30 officers, was lucky to get 6 in less populated areas. The issue came from when regular men were chosen (as they seemed as good leaders) they resisted. Many of these miners and herders and businessmen alike, normal people, wanted to fight for their country but they were not prepared to accept responsibility for their friends lives by becoming NCO's or officers. Men would create oaths vowing never to accept a promotion;being a private was a thing of pride in Kitchner's Army.
Notes:
^[1] Lance Corporals were responsible for a section of 14 men and would be paid 1s.3d./day compared to a Privates 1s/day.
Further Reading
The First Day on the Somme, Martin Middlebrook
Tommy: The British Soldier on the Western Front, Richard Holmes.