Einstein's theory of General Relativity (which I am treating as distinct from his theory of Special Relativity, which was earlier) was published in 1915. It is important to note that this happened in the middle of a World War, which was — as you can imagine — a bit distracting to everyone. Unlike Special Relativity, General Relativity relies on extremely complex math, which makes it extremely inaccessible even today. So it is interesting to see which scientists initially took it up as something worth looking at; they tended to be extremely mathematically-oriented physicists or astronomers, which was not the norm in 1915.

Karl Schwarzschild was one such mathematical physicist/astronomer who managed, in 1915, to produce the first exact solutions to the equations, to Einstein's delight. What made Schwarzschild think they were worth spending his time on? We don't really know — he died from an illness contracted during the war just the next year. We do know that Schwarzschild had already, as early as 1900, been looking at non-Euclidean geometrical models, and this made him somewhat unusual and perhaps "primed" for finding Einstein's work interesting.

More interestingly, though, is that Einstein's own work yielded three experimentally testable predictions that would differentiate it from Newtonian dynamics. One was a correct accounting of Mercury's perihelion, which would have gotten the attention of many astronomers (Mercury's perihelion was an annoying "anomaly" that had even had some astronomers proposing that there was an additional, tiny planet — Vulcan — that accounted for it), but by itself might not have convinced skeptics to drop Newton (a skeptic could argue that Einstein had built his theory around answering the anomaly, since it was well-known — this isn't true, but Einstein was aware of it and did use it as a "test case" for his evolving work). There was also a gravitational redshift, but this was very difficult to test.

And there was also the famous case of the gravitationally-induced bending of light, which Einstein had discussed as early as 1911 (with the wrong values), but by 1915 had clarified. Attempts to measure it after his 1911 prediction had not been successful (for fairly mundane reasons... like the outbreak of war); because the observations need to be done during a total eclipse of the Sun, they are not that easy to make.

So if you are looking at the period of 1915 to 1919, you can see why people might say, "well, that's interesting math you've got there, Einstein, but I'm not sure I'm willing to throw out everything I know so far to adopt it, without a little more reason to do so." This is an entirely unsurprising and frankly entirely justified response to a new approach that is not just a new refinement, but a wholly different worldview (a paradigm, in the term of Thomas Kuhn). People don't switch to a new paradigm unless it looks really promising compared to the previous one, and GR wasn't quite that, yet. And, again, it didn't help that a) there were plenty of other distractions (including, for example, the German and British scientific communities essentially declaring each other to be war mongers and frauds, as part of the war fervor), and b) the math was not the sort of thing most astronomers were trained to deal with.

The big change, of course, was the Eddington expedition in 1919, which finally got good eclipse photographs and thus could confirm that the light was being distorted as Einstein predicted. Even those results were tricky to parse, but let's just ignore that for the moment and focus on the goals of the expedition and its response. The world war had finally ended. Eddington was a British Quaker who raised money in order to make the expedition specifically in order to show that science could survive war — that a British Quaker could sail half-way around the world to try and prove or disprove the theory of a German Jew seemed like a very eloquent way to make that point. It would, as Eddington put it, "heal the wounds of war."

So while it was a scientific expedition, it was also a publicity expedition. Eddington deliberately wanted the world, especially the scientific world, to think this was a big deal. So when he confirmed that Einstein's theory seemed to work out experimentally, he made sure that it was carried by the newspapers of the world, that it was maximally sensationalized ("Newton overthrown"), that it was seen as a huge triumph. This is when Einstein became famous outside of the scientific community for the first time; his earlier work on Special Relativity had attracted some attention within the community (thanks, in part, because of its endorsement by Max Planck), but it was the eclipse that made Einstein a public figure (outside of Germany, anyway; within Germany he was perhaps somewhat known for his political activity as well, such as refusing to take part in the anti-British efforts and being a pacifist).

Were physicists and astronomers all made Einsteinians on account of these results, though? Not at all — the experimental data was pretty hard to parse (it was a lot trickier to take those eclipse photographs than is perhaps obvious today; their equipment was much cruder than we'd use today), and the observed differences are tiny-enough that your skeptical astronomer or physicist could say, "well, maybe, but let's wait for more data to come in." This is, I would point out, a common thing in science, and not a bad thing, up to a point — conservatism in adopting new theories or even accepting counterintuitive data insures that quirks don't become orthodoxy. (There are plenty of times, including in our present world, in which one study gives anomalous results that are later attributed to error or fraud.)

But what you get after 1918 is a lot of people starting to take Einstein's work very seriously — for both good and bad. You get the scientists, especially young scientists, taking up the challenge of learning the ins-and-outs of the equations, and expanding and extending Einstein's work to cover new situations. You have experimentalists wondering if you could prove other predictions that come out of the equations. You have a new generation thinking, "this might be worth looking into," because it suddenly looks more promising than the old theories at generating new work. It is of note that British astronomers in general were particularly interested in Einstein's work. Why? Because they could do the math, because in the UK the tripos system made it so that multi-variate calculus was a standard skill for all educated people (including astronomers), unlike practically anywhere else.

But you don't tend to convert the "old guard," ever. As Max Planck famously put it, you basically just wait for them to retire and die, and the new people to take their place. But interestingly with Einstein, you also get strong anti-Einstein movements, both popular and scientific. Einstein's work provoked people, including people who could never do the math. As it was taken up with great popular interest, and people started to read all sorts of political and cultural implications about "relativity" and "relativism" (none of which Einstein intended, but it doesn't matter), this meant that especially people who were on the conservative side of the political spectrum felt attacked by it.

So especially in Germany, but elsewhere as well, you start to see the harsh anti-Relativists organizing against Einstein, with rallies and books and denunciations. And even some very important German experimental physicists (notably Phillip Lenard and Johannes Stark, both Nobel Prize winners) took up this cause. These sorts of people also ended up being attracted to Nazism very early on, because it seemed to reaffirm their sense that their way of life was being attacked by bourgeois, cosmopolitan forces — and the fact that Einstein was Jewish played a big role in this as well. Stark and Lenard would, once the Nazis took power, attempt to parlay this sentiment into an officially anti-Einsteinian, anti-quantum physics movement called "Aryan Physics" (to contrast with "Jewish Physics"), and had a little limited success before the Nazis got tired of them and abandoned them (the Nazi Party, as it turns out, didn't care about abstract physics debates — and once the war started, they realized that the young physicists were the ones who would be the ones who were going to make contributions to the war effort, not these cranks). Stark, as an aside, barely avoided getting sent to a concentration camp, as the Nazis found him incredibly annoying (and thought he had a nice house that they might appropriate).