Wow, a question I can actually answer! In short, the concept of a global "Anglo-Saxon" union was somewhat popular among a small subset of the British Imperial Federation movement. I would absolutely not call it a popular idea, given how impractical it was (especially at the time), but some members of the movement likely saw it as a vague, future dream.

Proponents of Imperial Federation sought to create a federal British Empire, where the dominion parliaments would act as a sort of state/provincial legislature, with supreme power being focused on the Imperial Parliament in London. Most of these federal schemes advocating for uniting only the "white" dominions (Canada, Australia, New Zealand, and South Africa), as their linguistic homogeneity and similar racial background (overlooking all their non-British and especially non-white inhabitants) would ensure Britain's global superiority. I did my undergraduate thesis on the Round Table Movement, a later subset of the Imperial Federationists, so I have a bit of background knowledge on the topic.

The Imperial Federationist idea did gain some popularity in the last decades of the nineteenth century, as the British establishment became increasingly concerned about the relative decline of the United Kingdom compared to the far more populous and resource-rich Americans and Germans. As well, this is the era of linguistic pan-nationalism, where long-divided nations like the Germans and Italians were unifying. To Imperial Federationists, they were simply uniting the worldwide Anglo-Saxon diaspora. The short-lived but ambitious Imperial Federation League was formed in 1884 to lobby for Imperial Union, though while it found support from some members of the Conservative Party it dissolved not long afterward due to the sheer impracticality of its mission. Some Liberals, particularly the Liberal Unionists who would later defect to the Conservatives over Irish Home Rule, also supported the idea (in theory), most notably the wildly influential Cabinet Minister Joseph Chamberlain. In reality, none of these ideas ever came to fruition, largely due to the immense political and economic cost of the scheme.

Generally, most Imperial Federationists limited their vision to the existing Empire. However, a small fringe advocated for the union of all "Anglo-Saxon" nations, united by their racial heritage, language, and similar legal system into a dominant super-state. One such proponent was the Liberal MP Charles Dilke, who in the 1860s wrote a book Greater Britain his travels through the British Empire and the United States, filled with frequent musings about the similarities of the two cultures and his hope they would one day unite. Another is the infamous Cecil Rhodes, who included Americans in the Rhodes Scholarship in the hope that the American elite would be exposed to British culture and bring their newfound Anglophillia back to the United States. As Rhodes' Last Will and Testament Stated, “I … desire to encourage and foster an appreciation of the advantages which I implicitly believe will result from the union of the English-speaking peoples throughout the world...". Likely, Conan Doyle (a noted fanboy of the British Empire) fit into this camp, and was musing through Holmes about just how great an Anglo-American union would be.

If you're looking for an academic work that touches on this, I would check out Duncan Bell's The Idea of Greater Britain: Empire and the Future of the World Order 1860-1900, which is a pretty detailed (if a little dry) summary on the various schemes and concepts that Imperial unionists came up with in the late nineteenth/early twentieth centuries. If you want to get really into it (and if your local university library has copies), I would check out Dilke's Greater Britain or J R Seeley's The Expansion of England, which was (at the time) a wildly successful and influential look into British Imperial history which the author used in part to argue for Imperial Union.