What sort of situation would need to arise so that eventually, during the 1880s or 1890s, a war breaks out between the British Empire and the United States? What would be the likely outcome of such a war?
I'm not sure what potential triggers there would be at this time,
but as for outcome, I'd expect either a British victory or a white peace. The US armed forces were pretty tiny and underfunded for most of the nineteenth century, and whilst the situation did get better during the 1880s and 1890s, they were still pretty small by Great Power standards. OTOH, Britain managed to get hundreds of thousands of men to South Africa within a few months of the Boer War breaking out, and I expect they'd put in at least as much effort to defend Canada as they did to conquer SA.
Venezuela, Alaska.
Britain had the edge in naval power but a land war would've been different. And it wasn't just a matter of facing a big US Army at the end of a long logistical tether. How might France and Germany etc react? Could they take advantage of British preoccupation with the US to encroach on British possessions elsewhere?
At the time of the Venezuela crisis, Salisbury's cabinet counseled caution and compromise, which, I assume, mirrored the potential hazards.
Venezuela Crisis in 1895/1896, excellent TL here.
Big US Army? The US Army was down to around 25,000-27,000 officers and men by the time of the Venezuela crisis, when they mobilised the States Militia/National Guard for OTL's Spanish-American War a quarter of the men declined to serve and another quarter proved unfit to fight. You might want to note it took the Army a good couple of months to field a single corps based mostly around regular regiments and that was enabled in part by the fact a big portion of the logistics was handled by the Navy. A little while later they sent a second expeditionary corps comprised of mostly National Guard units but again transported by the Navy to the Philippines.
You might want to look at the numbers collated by this site for a broad idea http://alternatewars.com/BBOW/Stats/US_Mil_Manpower_1789-1997.htm
By contrast the British would start by sending a corps of 40,000 men to Canada with another corps following over about three months. They did this for the South African/Boer crisis which was happening twice as far away and even then managed to support that army in operations deep inland over largely roadless terrain. If things did not otherwise go according to plan the mobilisation at least worked straight from the text book.
Unlike the Americans the British had an actual trained and equipped logistics branch (in fact being the British they had two the Army Service Corps and the Army Ordnance Corps) while the US Army relied on officers and sergeant on secondment from their regiments and would have started the war with a desperate shortage of wagons for road moves.
The image of the vast American horde army Zergling rushing Canada is a bit fanciful. Things might get complicated on the ground but at the end of the 19th Century the British had good cause to think they could hold Canada if push comes to shove.
Of course, alot depends on just what the conflict breaks out over and if the US has a ramping up period to get formal military units organized in the relative ease of peacetime with the expectation of a war.
Not saying this doesn't contain many good points, but here's a few caveats that also need to be included to make a Boer War comparison fair. The Boers had no large manpower pool and industrial base to "ramp up" from. The front was singular and significantly smaller, thus allowing the British advantage in being able to concentrate firepower to come into play and maximize the effect og on the ground Pioneer activity and deploying logistical resources. The Boers had no means to disrupt British commerce. South Africa was widely aknowledged to be part of Britain's sphere by the international order and thus their actions carried little risk of diplomatic repercussions
Of course, alot depends on just what the conflict breaks out over and if the US has a ramping up period to get formal military units organized in the relative ease of peacetime with the expectation of a war.