It’s impossible. Or perhaps a better way to say that is: It's technically extremely unlikely, not to mention there is no primary source evidence to even hint at pre-european ascents of any 7000m+ Himalayan peaks, let alone Everest.
Wade Davis’ excellent book Into the Silence does an excellent job of addressing this question, but I will lean on some personal knowledge here as an active mountaineer as well. It’s worth noting at the outset the Sherpa are one of several ethnic groups based near Everest and are famed for being effective exporters and more recently guides and alpinists.
The strongest evidence for the lack of ascent is interactions with locals on the first attempts at just reaching Everest's base. The local population just couldn’t wrap their heads around what the first expeditions were trying to accomplish. There is no oral tradition of pursuing peaks for any reason. Climbing high into terrible weather, avalanches and seemingly certain death for not tangible result did make any sense. There were traditions locally in which monks will go higher to caves to meditate. These aesthetic monks though were far from the norm, and rarely ventured higher than 6000m. The initial assumptions was the Europeans were prospectors or spies. The idea of doing any thing like this for scientific or recreational purposes was not relevant to their practices. This in itself suggests no communal level interest or understanding. Today we take it for granted, but even backpacking was not at all popular or socially understood well in the late 1930s even in the West.
The main challenge for the early expeditions is that it was exceedingly difficult to find the base of the mountain. There was no known route to even reach the base of the mountain. Guides hired to assist didn’t actually know the area, though some pretended to know hoping to earn some income. The reason appears to be that there was no hunting or grazing in either or the two areas used to establish base camps. Exploring these areas required ascending treacherous moraines, and wandering along glaciers, with only the prospect of more ice. Without any utility these spaces served no purpose. Life at these altitudes is harsh, growing seasons short, and there was not time to build skills for recreation when his subsisting was so taxing. Meanwhile the British expedition had trained for decades in the alps with specialty equipped and were well fed, motivated and fit. In other words they had reason and ability.
But let’s dive into the climbs from practical terms.
On a physiological level there is very little likelihood that a Sherpa,or other Nepalese persons could have ascended the peak. The second and third reconnaissance’s in 1922 and 1924 demonstrated that oxygen assistance was necessary. They failed to push past 7600 until oxygen was used, allowing for the first climb past 8000m. And this was for teams of the most experienced climbers, using the lightest and most effective equipment of the era. One of the second ascentionists (with oxygen) was a Sherpa, but guided by a Kiwi. While the Nepalese were more acclimatized, they had diets and a general lack of nutrition that evened things out.
They also lacked the technical climbing skill required to bypass the challenges of the mountain. The south col route that is the easiest first requires passing through the Khumbu icefall. It's a jumbled maze of seracs ( towering chunks of detached glacial ice). Today ladders are used to cross most of these sections. Crampons are also necessary to keep purchase in the ice and snow. The 1921 expedition used hobnailed boots as crampons had not been fully realized. They would chop steps with ice axes to ascend. There are examples of stone age crampons so this at least is feasible, but the technique required to actually use these tools was not recorded in any peoples of these areas. ( See above on motivation to climb) The early expeditions struggled immensely to keep the porters going as they had no mountain craft skills or abilities, and lacked the physical fitness to do much of the work (unlike today where the opposite is true). There were a few exceptions but all were deeply inexperienced and thrived once shown techniques perfected in the alps.
These early expeditions launched the career of porters for many Nepalese and Sherpa and as more Europeans flocked the valley changed dramatically. But these changes are post European arrival, as the Europeans created a lucrative career than enabled healthier lifestyles and communities built around this burgeoning industry. It also built traditions that made later expeditions possible. The First ascent of Everest was with a Sherpa for good reason.
Lastly there is a logistical challenge that pre-european mountaineers would have had to overcome. Mountaineering if this type requires setting up and stocking numerous camps over a period of a month.Even today with all our hyper light gear most ascents share more in common with siege warfare than alpine style climbing. These camps require specialty stoves to melt snow into water as it’s too cold for carried water to relief upon. High calorie food is required to counteract the appetite suppression of high altitude. Tents need to be brought for shelter to rest. These have to be light but strong enough to last through intense storms. It’s takes almost a full month to acclimatize and stage gear to make this all possible today with folks in incredible shape, with athleticism on possible due to modern living and diets. This kind of logistical complexity was seldom observed in this area. When it was, it generally revolved around religious festivals or moving officials through complex terrain of the Himalayas.
For a slightly malnourished person, with no equipment or supply train to support and provision them to have made it to the summit, with no ice axe, crampons, or high altitude clothing is impossible. If there was a shred of oral history or archeological evidence it might be worth pursuing. But there is not.
Edit: A second source I’m using for this is the superb Mountaineers: Great Tales of Bravery and Conquest published by the Smithsonian. It’s one of the more complete texts covering the entirety of mountaineering history from Otzi to today.
While more can always be said, you may find this answer from /u/caitrona useful:
It's not really a historical question, rather a technical climbing question and a cultural question. The answer, I would say, is "not very".
Broadening the technical question to "were there earlier successful attempts on any route", the only two routes which are not fiercely technical are the South East Ridge from Nepal and the North Ridge from Tibet.
Both nonetheless have substantial technical cruxes. Before the 2015 earthquake the South East Ridge would involve the Hilary Step, and the North Ridge route has, in particular, the second step.
The Hilary step was not a major technical problem but was nonetheless a significant piece of climbing at altitude with 1950s equipment; subsequent attempts mostly used fixed ropes. The second step has been climbed on-sight and unaided in recent years but was graded 5.10, and most ascents used a fixed ladder placed in the 1960s.
Although it would be romantic had Mallory and Irvine climbed the second step in the 1930s, there's very limited evidence suggesting they did, and given the equipment they had a lot of reasons to believe they didn't. Even if you assume, which is something of a simplification, that the high-altitude fitness of local people removes the issue of oxygen, the second step is a technical climbing challenge which would be difficult in the 19th century even were they at sea level.
The cultural question would be whether anyone would make the attempt. Leaving aside religious objections, there is little point in doing it beyond "because it's there" and it's unimaginable that a very first attempt would succeed.
You're therefore left with the question of whether people would continuing killing themselves in pursuit of an entirely pointless end, to which the answer is most likely "no". Similarly, there is very little evidence of attempts on considerably simpler mountaineering problems like Mont Blanc until the late eighteenth century. Western European history is at least as complete as that of Tibet, and there no record, tradition or even myth of earlier ascents. The same goes, with caveats, for Himalayan high mountains: there is no folk memory or myth saying they were climbed.
The definitive book on Everest, or at least on failures to climb it, is Salkeld, 2003. The earlier Holzel and Salkeld, 1999 is specifically on Malory and Irvine and addresses in detail the difficulty of the North Face with the equipment of the 1920s, but it's mostly subsumed by the later book.
Salkeld, Audrey. Climbing Everest: Tales of Triumph and Tragedy on the World's Highest Mountain. National Geographic, 2003.
Holzel, Tom, and Audrey Salkeld. The mystery of Mallory and Irvine. Random House, 1999.