Elijah, when unleashing the toxic clouds to the Mojave Wasteland, would profoundly shift the trajectory, where the political conflict between NCR and Legion would be abruptly crushed. With Caesar's cardinal target inaccessible (or at the very least, extremely taxing to maintain), the Legion faction would be directly damaged; a venerated ideal of Caesar, the constructor of the quasi-Roman Empire, would diminish as the organisation would be suffering high casualties. If Caesar is still persistent with his ultimate goal of securing the Dam, I envision that the Legion, weakened and crippled, would expedite their demise, spending a vast congregation to their untimely deaths. Interestingly, if the hubristic disposition of Caesar would concede to adaption (the degree, however, remains difficult to address), the Legion could decide to enhance their territorial aspiration to the East. I could even imagine Caesar developing propaganda about the mystical clouds as a justification to change the direction of the Legion. Therefore, it results in the organisation in assimilating greater tribal groups, so the relative stability could remain intact - though the notion of if the Legion would persist after Caesar's death is highly debatable, even in a timeline where the Legion decided to expand East.
As for the NCR, meanwhile, the obstacles placed by Elijah would nullify direct intervention in the Mojave, a campaign which Kimball's administration invested voluminous amounts of money, time and lives. Since a significant quantity of civilians would be castigated by Elijah in the aftermath, political and social instability could potentially blossom rapidly, where the rudderless and inertial state of the government would be underscored. To the extreme side of the equation, with a society disgruntled and traumatised by the events of the Mojave, radical thinking could involve a takeover in the political ideology of the Republic. Hence, it echoes growing resentment to the upper echelon. In terms of foreign policy, the NCR could circumvent the Mojave (while it would be disappointing to no be able to acquire an economic nucleolus) and continue the expansion with relative effectiveness.
Leading directly to the nadir crisis of the Mojave Chapter, the plunging of morale and the macabre disposition of the once relatively capable organisation has soured Elijah. Conversely, however, there are pockets of individuals who affiliate themselves with Elijah, especially when compared to the perceived isolated, impotent leader of McNamara. Nonetheless, whether or not the BoS would accept Elijah as the conquering hero remains the issue of defeat. If yes, the BoS would be the only organisation that would remain in the Mojave (they are situated in the bunker), thus would become the automatic dominant faction. If no, the entire concern for the toxic gas would be completely futile - well, it would satisfy Elijah's obsession to destroy the NCR, despite how he was the driving factor to the BoS's demise. Alternatively, perhaps even civil war ignites to contest over the orthodox leader of the BoS.
Above all, the Mojave would be a graveyard - the population would plummet, as too would the economy and production. Validating the encumbered ego, Elijah would be the ruler of nothing; everything would be macabre and lifeless, symbolically exhibiting the cruel traits of Elijah.