Short answer: A long time, and incompletely even then. 1930s if you want a decade for real consensus, 1860s for initial important impact.

Somewhat longer: Darwin published Origin of Species in 1859, and had some early adherents. He didn't do an enormous amount of proselytizing himself, but others took on that role (Thomas Henry Huxley, most notably). One problem with answering this is what's meant by 'Darwin's theory,' since there was a lot of larger scientific and semi-scientific debates that Darwin was contributing towards, and Origin of Species was important but not definitive for really any of them.

There were serious debates about how old the Earth was, for example, with some people saying very very old (approximating what we believe/know today) based on things like sediment from rivers, but others (for example, Kelvin) arguing it must be younger because if you calculate the energy the earth receives from the sun and the average temperatures, it can only be a few thousand years old - legitimate science, and only wrong because no one knew about all the extra energy slowly released into the earth by radioactivity. Then there are of course the big issues of biblical interpretation, which mattered a lot to many scientists as well as everyone else. Some influential, important scientists never accepted human evolution (which was a point Darwin was a little careful about early on anyway).

Really, Darwin's work wasn't accepted as mainstream science in the same sense we accept something like General Relativity until the Modern Evolutionary Synthesis hit the stage in the 1930s-40s. Until then, there were lots of variations on evolution that had more or less to do with Darwin.

Meanwhile, Darwin had a big cultural impact outside of science in terms of things like 'social darwinism,' where the idea of 'survival of the fittest' was taken to mean that those in power must deserve it, and any form of charity of aiding those with less power would just weaken the species. This also tied into eugenics, which was a serious and important branch of science and culture in the early twentieth century.

Recommended reading: Bower, Evolution: History of an Idea; Numbers, Darwinism Comes to America; Larson, Summer for the Gods: The Scopes Trial and America's Continuing Debate over Religion and Science