Main questions:
When & why did enthusiasm for Georgism fade?
Is there consensus on the performance of the Georgist experiments that were tried? Did they "work"?
What's the closest a Georgist "single tax" ever got to adoption at the national level in any country?
Some background:
Popular book
Here's Wikipedia on how popular the book was:
Jacob Riis, for example, explicitly marks the beginning of the Progressive Era awakening as 1879 because of the date of this publication. The Princeton historian Eric F. Goldman wrote this about the influence of Progress and Poverty:
For some years prior to 1952 I was working on a history of American reform and over and over again my research ran into this fact: an enormous number of men and women, strikingly different people, men and women who were to lead 20th century America in a dozen fields of humane activity, wrote or told someone that their whole thinking had been redirected by reading Progress and Poverty in their formative years. In this respect no other book came anywhere near comparable influence.
Progress and Poverty had perhaps even a larger impact around the world, in places such as Denmark, the United Kingdom, Australia, and New Zealand, where George's influence was enormous. Contemporary sources and historians claim that in the United Kingdom, a vast majority of both socialist and classical liberal activists could trace their ideological development to Henry George. George's popularity was more than a passing phase; even by 1906, a survey of British parliamentarians revealed that the American author's writing was more popular than Walter Scott, John Stuart Mill, and William Shakespeare. In 1933, John Dewey estimated that Progress and Poverty "had a wider distribution than almost all other books on political economy put together."
Wikipedia lists major figures who said they were influenced by the book, or who said they agreed with the book's central recommendation of a confiscatory tax on the rental value of unimproved land, and it's a who's who of the 20th century: Albert Einstein, Helen Keller, Winston Churchill, Clarence Darrow, Leo Tolstoy, John Dewey, Bertrand Russell, Aldous Huxley, Friedrich Hayek, HG Wells, George Bernard Shaw, Milton Friedman, Emma Lazarus, John Haynes Holmes, and more.
Limited implementation
Wikipedia's article on land value taxation says LVTs have been implemented in many places. But the recommendation in the book isn't just an LVT--it's an LVT that replaces all other taxes, especially those on labor and capital. The whole argument is that taxes on labor and capital (as well as tariffs, which are taxes on international trade) are harmful while an LVT is not just harmless but a positive social good, so an LVT should be maximized while the others should be zeroed out.
Here's Wikipedia on communities in which Georgism has been tried. It lists a couple towns in the US. Fascinatingly, it also lists a German-controlled territory in what is now China's Shandong Province. (Any information on this episode would be highly appreciated!)
But nothing at the national level anywhere, as far as I can tell.
First of all, it's worth noting that Georgist ideas have not entirely faded from view; certain notable economists (such as Nobel-winner Joseph Stiglitz) claim George as an influence, and his work has been mentioned in mainstream publications over the last few years (such as this 2019 article in the Atlantic). However, in general, the progressive tide simply changed focus. This is largely due to the increasing prominence of the socialist movement in the late-19th and early-20th centuries, pushed Georgism to the side somewhat. From then on, it became more of a background influence; people across the political spectrum (from Milton Friedman to the aforementioned Joseph Stiglitz) continue to reference George's ideas, but there aren't a lot of people who would outright call themselves "Georgists" anymore.