Darwin was largely accepted very quickly. Biologists understood what he was saying and saw the power of the theory.

The biggest scientific objection dealt with time. Before Darwin was known that the Earth was old but scientists were not sure how old. Agassiz's work in the early 19th century on ice ages show the world was at least hundreds of thousands of years old. With the understanding of thermodynamics, but lacking knowledge of radioactivity, physicists in the last 19th century figure the Earth was at most 100M years old (other wise the core would have cooled off and no volcanoes). Common Descent required an Earth of billions of years. The discovery of radioactivity pushed back the physics estimate and was a major win for biology over physics.

The late 19th/early 20th century saw a lull in use of evolution. They did not know how inheritance worked and that made it harder to use the theory. It looked like inheritance was blended rather than particulate. If so then descent would not work the way he stated. With blended inheritance an offspring gets the "average" of the characteristics of the parent. With particulate you get one trait or the other. It was not until the 1930s that they were able to show that inheritance was particulate.

The development of population genetics with Fischer, Wright, and Haldane in the 1920s brought Darwin back to the spotlight. And the Modern Synthesis in the 1930s and 40s pretty much cemented it in place.