Useful stats--in 1870, the US economy was 98% the size of Britain's itself, or around 35-40% compared with the British Empire as a whole. That's larger than either Germany or France. In 1913 it had grown by 527% to be the largest in the world. If we assume 70% growth because of a bigger military and less foreign investment, than in 1913 it would still be larger than Germany and (Metropolitan) France combined as well as a sizable fraction of the British Empire.
You don't realise how international diplomacy and great power relations worked back then. It was less diplomacy and more two mafioso bosses staring at each other and seeing who'll blink firsy, and the results were completely and utterly bonkers.
The US and Britain nearly went to war over a shot pig. The British sent 13 000 soldiers, 26 000 carriers and animal handlers, 40 000 draught anials, inlcuding 44 elephants to trek all over Ethiopia to liberate about a dozen hostages, only half of which were British subjects that Enperor Thewodros had taken 1868.
Every country could see what happened to countries that ignored challenges to their territory, prestige and power. The Ottomans and Qing were being picked apart. Tibal Confederacies and Khanates were goaded into splendid little wars and annexed, And the fate of Poland-Lithuania was still fresh in the mind of Europe's rulers. Anyone who looked weak would have to face ten times worse as the vultures started circling. And they KNEW that.
Thus every attempt to take territory will be met with a balls-to-the-walls insane reply with everything that country got. Look how the Spaniards, despite being outmatched from the start, not even in control of Cuba and having not a single chance still sailed their entire navy across the Atlantic to have it murdered by the US at the Battle of Santiago Bay.
So, "why should they over a few islands" is to completely misunderstand how diplomacy and international relations worked in the era.
I think you misread my post. I was saying that if Britain were losing a war to the United States in the 19th century, the rest of the world isn't going to demand them to stop and hand back those Caribbean islands because the United States is not Meiji Japan and the Caribbean is not a highly strategic area like China for trade, warm water ports, etc. If we assume the US ever had gone to war with Britain in this scenario, then this wouldn't be their debut on the international stage (likely involving throwing their weight around diplomatically or militarily against Spain and Latin American nations). It would not be a minor power appearing out of nowhere and interfering in the affairs of major powers as Japan was after the First Sino-Japanese War, but a nation viewed as a rising power having overcame the world power.
The British did not invest in geopolitical rivals. Not in Russia before the Triple Entente (and hardly even after that, the French handled most of that), not in Italy before it left th e Central Powers and so on. British banks were pretty good at following the lead of the government. Building up to become a rival to Britain would have to be motivated internally, and the only version I see working is some kind of jimgoism against Britain, which will lead to incidents, diplomatic, civilian and militarily. Regardless whether these incidents are handled well or not, the US buildup will nto happen in a dark vacuum. Britain will notice and respond, and you'll have a western hemisphere arms race, and in that environment, British banks will not be ledning money in the US, and British investors, fearing having their property seized in a potential war or trade incident, will not invest.
I think a better comparison would be the UK in regards to Imperial Germany, where British and German firms engaged in cooperation in sharing advances in engineering and both countries enjoyed a brisk trade.
This also ignores the limitations of a 19th century naval blockade. Even the small Confederate States Navy and their blockade runners still allowed their nation to carry on some trade internationally. Blockading two coasts against a far larger navy and one with a huge amount of merchant shipping and experienced sailors would be far more difficult. Britain's ally Japan in this era also does not have the ships or projection power necessary, especially before the First Sino-Japanese War. So in the vast Pacific, at least some trade with China and Russia could still be done.
During WW1, some 1,500,000 men served in the Indian army, taking an accumulated 120,000 casualties. It's total contribution to the war effort exceeds that of all GB's Anglophone colonies
combined. (Sauce:
https://encyclopedia.1914-1918-online.net/article/war_losses_india)
Germany was not the only central power though, and the Indian and African soldiers were vital for the colonial war effort. And since WW1 was a
World War, as in
world-wide, these soldiers were absolutely indispensible to the Empire.
Colonial soldiers were generally pretty loyal to the Empire so long as they served outside their nation of origin. They were mostly volunteers from middle-class families that were compensated handsomely by the Empire. And many of them, Sikhs and Gurkas for instance, would come from a long tradition of military service that absolutely forbade desertion. Even when there were revolts, the British policy of playing different ethnic groups against each other ensured that oftentimes, these soldiers had no qualms fighting rebels in the name of the Empire.
Besides, where would you go as an Indian deserter in North America? The Americans'd just lynch you on the spot, most likely.
Most of them were far closer to home in the Middle East or Africa, and according to your link includes 563,369 non-combatants, post-war interventions in Afghanistan, Egypt, and Iraq. According to that, less than 10% of that number served on the Western Front in France. It also lists the disparities in treatment Indian soldiers received, especially dependent on theater.
Germany was the lynchpin behind the Central Powers in World War I, and if we're going to assume a World War, then Indian soldiers may well be needed elsewhere than the United States. A world war in this era with a strong American military would likely be pretty disastrous for an Anglo-French alliance, especially in the 1880s when Russia will be at worst neutral toward Germany, Italy on Germany's side, and Japan still limited in power projection.
Even if they can only cripple American shipping in the Eastern hemisphere, that's a huge blow to the United States. While exports are not a vital part of the economy for the US, quite a few people make their money in the shipping industry, and the collapse of that industry would have severe consequences on American society and politics.
If they take the Pacific, they can also dump that huge Indian army I mentioned right onto California's shores, which would need to draw any supplies and reinforcements by train from the East Coast, and has a vast shoreline that'd require millions of men to control effectively.
Taking a train from the East Coast is a hell of a lot easier than taking a steamship across the Pacific, especially since the Pacific is so vast that in the age before radio it would be very difficult to effectively patrol. There is also a scarcity of British-controlled ports in the Pacific. There are neutral Latin American ports, there's a cluster of ports like Esquimalt (the OTL base) and Vancouver, and everything north of that is minor at best. British Columbia is too remote and sparsely populated pre-1900 relatively to the US population on the West Coast and there's a sizable number of American citizens there. Vancouver Island would be difficult to take entirely due to British strength and likely coastal forts, but it could be neutralised via use of sabotage, small minelayers, etc. The Salish Sea is a maze of islands meaning lots of places for torpedo boats or armed merchant raiders to hide (this applies to Britain as well, but the US has more local shipbuilding capacity).
And there are the Mexicans to consider, who'd jump at the opportunity to get back some of their lost empire, especially if they are backed by the largest military power in human history up until that point.
Porfirio Diaz and his regime were owned by the United States government and American corporations, so I don't see why this would change. Like in both World Wars, Latin American nations benefit greatly by staying neutral.
And you're assuming a rapid and easy victory in Canada - which is possible, but a bloody quagmire is too, especially if the British are aware of America's huge military buildup along the border (which they certainly would).
France has had a special relationship with the US since forever (first two Republics in the West since Rome and all that), but a unilateral invasion of Canada is not something they can ignore if they want British aid against Germany. At best, they'd be a moderating influence on the British. Spain'd go to war for the simple reason of eliminating America as a threat to the Carribean, and the prestige of winning a war for once.
But the US could counter that by offering their own aid--given they'd have a large arms industry, still be a major exporter of grain and oil, and have either the third or second largest navy, that's a very good offer. The US also doesn't care what France does in Africa and would help them negotiate new borders or in a pinch, even readjust Liberia's borders (as that nation was wholly dependent on foreign loans). And in the 1880s, there were more cordial relations between France and Germany over fear of a colonial war with Britain, and these fears remained until the early years of the 20th century.
