This isn't a direct answer, but some basic information on Britain and America at the end of the 19th Century might clarify why Sir Arthur Conan Doyle probably longed for such a reunion.

At the end of the 19th century, Great Britain's empire was very nearly at its greatest extent, but the nation was nonetheless also obviously in decline. The United States's GDP passed GB's around 1870, and the newly unified Germany's did so shortly thereafter. Moreover, Russia seemed to be on the way up.

Many observers at the end of the 19th century saw (correctly, as it turned out) the 20th as likely to be shaped by two (or possibly three) giants: the US, Russia, and possibly Germany. A re-unified English-speaking world might thus have seemed an extremely attractive idea to an strongly nationalistic Brit watching his nation's star eclipsed by America's. It would have resulted in an enormous nation, massively strong economically and militarily, and thereby assuaged any fears of being surpassed by the other rising powers.

Discontentment under the Articles of Confederation was so great that had the Consitution never been ratified, Hamilton feared a possible reunification with Britain.[1] My high school history teacher, a brilliant and dedicated historian, also underscored reunification was on some newly independent Americans' minds under the failings of the Articles, but I cannot find a detailed source.

[1]. Conjectures of the Constitution, a copy available here

In short, no.

During the late 19th century, it was clear that America was the superpower of the future. Gladstone in 1879 said of America: "It is she alone who, at the coming time, can and probably will wrest from us our commercial supremacy...We have no more title against her than Venice or Genoa or Holland has had against us".

With this increase in American power, British and American interests inevitably clashed. In 1895 America intervened in an Anglo-Venezuelan dispute and sent an ultimatum to Britain to submit the dispute to arbitration or face war. Britain backed down. Not long after, America ensured that the Panama Canal was in her hands, in effect making the passage of the Royal Navy between the Atlantic and the Pacific dependent on American goodwill.

In 1898 Britain yielded to American pressure over the disputed border of Alaska/Canada to submit the dispute to arbitration, which mostly favoured America. The Boer War shortly after inflamed American opinion against Britain. In 1900 the annexation of Canada narrowly failed to be adopted by the Republicans.

However the opinion in Britain was pro-American. The British ruling class in the late 19th century had seen an influx of American heiresses (130 by 1914 in the aristocracy) and the common image/myth of America in the British mind was of a kindred Anglo-Saxon country. Americans were Britain's 'cousins' according to this view, despite the fact that the English-descended population of American had fallen to one third by the late nineteenth century. Lord Bryce's book The American Commonwealth, though fairly documenting all those aspects of American life that rendered her a foreign rather than a cousinly nation, came to be heavily influential in shaping this view. The British were misled by race and class (their affinity with WASPs on the Eastern sea board) into viewing America as just one step removed from, for example, Canada and Australia. The Americans held no such view.

There widespread notions of affinity to America as a kindred nation led to a few British imperialists to dream of an Anglo-American union but opinion in America saw Britain as a rival foreign power, without any sentimentalism. So I don't think, looking over Anglo-American relations in the late 19th century, was there any chance of a political union between the two states.