Having a Proper Sikh Wank
Well, I've been working on my own timeline based around this scenario for some time, and conducted a fair bit of research. In my timeline's POD, the capable heir to the Sikh throne, Nau Nihal Singh, escapes his freakish death (or, rather, opportunistic assassination) IOTL, avoiding injury in the archway collapse at the Lahore Fort upon his return from his father's funeral. IOTL, eyewitnesses asserted that the collapse only knocked Nau Nihal Singh unconscious, with few visible signs of injury; the crown prince was subsequently carried inside the fort by members of the Dogra faction and carried back out dead, with his skull smashed in.
In contrast, ITTL, Nau Nihal Singh survives to become Maharajah Nihal Singh, and subsequent Dogra attempts to kill him and cause the termination of his unborn child through slow poisoning (the same method used on his father Maharajah Kharak Singh) are exposed before they can bear effect. The Dogras abscond Lahore, and a brief civil war ensues, culminating in the defeat of the Dogras and their allies in the Punjab Hill States, and the consolidation of power by the Sindhanwalia-backed Sikh Raj.
Of course, you'd assume the internal conflict would leave the Sikh Raj more vulnerable to British aggression during this period and in its immediate aftermath; but with the British East India Company's commitments in Afghanistan at this time (1840>1842) and their vital supply lines through the Punjab, running through regions which would be contested between both sides in a Sikh Civil War, the risk to the convoys would have effectively ruled out direct military intervention on either side's behalf, limiting their intervention to support through propaganda and the supply of arms. The B.E.I.C favour the potential divide-&-conquer opportunity offered by backing the Dogras; but without direct military intervention, it makes little difference to the outcome.
Worse, it proves fatal to diplomatic relations between the EIC and Sikh Raj, and stirs public sentiment in the (primarily non-Sikh) heartland of the Punjab against the British. At the inevitable outbreak of the 1st Anglo-Sikh War, in spite of the losses incurred over the course of the Sikh Civil War, the defending Sikh armies are no smaller than they were IOTL, comprised of far more seasoned troops, many of whom already gained experience facing the British weapons in the hands of Dogra troops- and far more importantly, ITTL they're actually led by commanders who want to win, unlike IOTL where the Dogra commanders Lal Singh and Tej Singh had a vested interest in losing their battles and the war, in order to secure Kashmir's independence as a Princely State under the direct rule of their family dynasty.
As High Plains Drifter pointed out, the Sikhs had real opportunities to win the 1st Anglo-Sikh War even IOTL, so a decisive victory for the Sikhs ITTL is pretty much a certainty. The outcome of a 2nd Anglo-Sikh War, in which the British would certainly learn lessons from the first conflict and commit a much larger, more well-equipped and well-led force, would be a far less comfortable one, but it's still more than feasible that there would be enough Sikh victories in battle to force a B.E.I.C withdrawal, and an effective Sikh victory in the conflict akin to that of Afghanistan in the 1st Anglo-Afghan War.
Of course, defeat in a 2nd Anglo-Sikh War wouldn't be likely to curb the B.E.I.C's ambitions of to extend its dominion over the Sikh Raj and effectively complete its conquest of India, or to regain the shattered prestige of the military forces in its employ. However, the concentration of troops in the build-up to the B.E.I.C's 3rd planned campaign against the Sikh Kingdom soon backfires in a big way, with the outbreak of the Great Indian Mutiny further strengthened in intensity by the greater numbers of Sepoys concentrated in the north-west of the British Raj, their resentment at being blamed for the B.E.I.C's losses in the previous Anglo-Sikh Wars, and the deep unpopular prospect of having to wage another one shortly.
Unlike IOTL, where the Sikhs directly intervened to help quell the uprising, the Sikh Raj provides every form of support short of direct military intervention, allowing the Mutiny to succeed in its objectives in the short term, seizing effective control of Haryana, Rajasthan, northern Madhya Pradesh and western Uttar Pradesh, and installing Bahadur Shah back to power. Unfortunately though, I can't see the fragmented Second Mughal Empire enduring for long; ITTL, it persists for seven years before splintering upon the death of Bahadur Shah II (2 years later than IOTL, in the royal court rather than in exile) allowing the British Raj to retake control, but the Sikhs have more than enough time to consolidate significant territorial gains.
West of the rebellion, many Princely States which have no desire to become part of a renewed Mughal Empire turn away from the folded B.E.I.C to become tributaries of the Sikh Kingdom, the strongest military power in the region; these include Kalat (Baluchistan), Khairpur (Upper Sindh), Kutch (N.W. Gujurat) and Bahawalpur. With the token British presence driven out of the Lower Sindh region by civil uprest, the Sikh Raj's territories are expanded to encompass the entirety of modern-day Pakistan, along with the modern Indian states of the Punjab, Kashmir and Himachal Pradesh. The Sikh Empire hasn't yet reached its maximum extent though; and in its enduring existence, it soon begins to have truly global repercussions...
p.s, Delvestius; the Sikh Raj only endured for 30 years or so, but in that time, the only period in the history of Sikhism where it was an ecclesiastical religion with the backing of a government, the Sikhs' share of the population in their kingdom rose from only 5% to around 19%, almost a fourfold increase. Even if that increase was linear rather than progressive, and taking into account the respective populations of the areas they'd be expanding into, you'd still see Sikhs holding at least a plurality in their own Kingdom ITTL by the end of the 19th century.
p.s.s. Another interesting thing to consider, for the early to mid 20th century part of such a timeline; in the period immediately before its conquest by the British, the estimated GDP, population and the size of the Sikh Raj's armed forces exceeded those of Japan by a huge margin. Without Britain siphoning off its trade wealth, with its own industrialisation initiatives (already being carried out in its military forces IOTL through the Fauj-i-Khas) and with the raw materials and manpower at the Sikh Raj's disposal, would the Sikh Raj or the Shogunate have been better placed to become the dominant power in Asia? Hmmm...
