Lastly, there is always at least one other power waiting in the wings to take a stab at the role. While the globocop might welcome a deputy, will the deputy eventually want to be the sheriff? And where does that leave the former sheriff? The British Empire had two such prospects in its shadow by the end of the 19th century; Imperial Germany and the United States. With America's victory against Spain, it became the defacto deputy to England, while Germany was still looking to break out of central Europe and increasingly saw war as its only option.
I think that this is an inaccurate characterization of a few things:
1. the role of the British Empire
2. the way German elites approached geopolitics in the early twentieth century, and
3. the reason the First World War broke out.
There's also a problem that your definition of "globocop" seems a bit fuzzy. (Also, I can't read "globocop" without thinking of "Globocnik".)
If the requirement is an ideology of national exceptionalism in a Great Power, then yes, the British Empire possessed that quality. So did the other contemporary Great Powers. You can easily argue that Britain found itself more willing and capable to participate in foreign interventions than other powers due to the nature and extent of its empire, but that's a difference of degree, not type.
If the requirement is an actual, concerted effort on the part of the most powerful nation to enforce peace in order to achieve prosperity, that is hard to substantiate. The nineteenth century saw its fair share of wars both in Europe and abroad, and the British Empire's role in preventing them, stopping them, and participating in them is highly uneven. Sometimes you argue that some of the wars that Britain
started were the ultimate expression of the British Empire's role as the enforcer of global peace, which is hard for me to swallow. Sometimes you argue that the wars that Britain failed to prevent (or even precipitated) weren't that big a deal. When you explain away the Austro-Prussian War - the conflict that saw one of the largest land battles in history up to that point and completely changed the history of Germany - as a "sideshow", it makes for an awfully mendacious read. Even the contemporary Britishers thought that the war made for one of the most "decisive" events in world history. It wasn't the only one; Disraeli described the subsequent Franco-German War as "revolutionary". The fact that the British Empire's role in resolving these wars was basically nil does not speak well for its supposed role as "globocop".
If the requirement is that the Royal Navy ensured global prosperity through trade, that is simply untrue. That global prosperity was instead based more on declining transport costs due to the application of steam power technology to sailing ships and railroads. It didn't have all that much to do with peace in and of itself. (Which is, in fact, one of the reasons the British Empire wasn't terribly interested in global peace in and of itself.) The prosperity
ended when war broke out, but that again didn't have much to do with the Royal Navy or keeping global trade safe. Apart from a very brief episode in 1917, piracy (i.e. something the Royal Navy could actually prevent) did not threaten global trade, but global wealth, per-capita income, productivity, and so on began to decline in 1914 and continued to do so for the next three decades.
So I have trouble understanding why you think that this "globocop" thing is a Thing.
Structurally, most historians of international relations describe the period after the Revolutions of 1848 as "multipolarity", not as the hegemony of a superpower. Before 1848, the so-called Congress system was in fact a dual hegemony exercised by Britain and Russia over Europe and involving limited competition outside of it. Afterwards, while British industrial and economic and naval power remained strong, and while "dual hegemony" with Russia ended in 1853, the actual record of British governments exercising that power in a unipolar way is almost nonexistent. At no point did Britain attempt to "lead" Europe diplomatically, nor did British governments openly oppose the rest of Europe. Instead, British leaders made infrequent combinations with various powers against other powers, much as they had done in the past, while mostly remaining aloof from Continental affairs (the so-called "Splendid Isolation"). Eventually, both Conservative and Liberal leaders understood that Britain lacked the military and economic power to survive in early twentieth-century politics without actual allies, so the British found them. The fact that individual Britons often treated these alliances and agreements with the sort of supercilious disdain that implied their superfluity does not mean that they were, in fact, superfluous. They were absolutely necessary, and they strongly argue against any unipolar superpower diplomatic structure. It's easy to argue that the British Empire was the most powerful country in the world by most metrics in 1900, but most historians tend to classify it as a first among equals rather than a power that tilted the entire gravity of world politics towards it.
I simply don't think that describing the British Empire in this way is particularly useful.