It was not accepted immediately, but quickly, courtesy of Bohr. It was he who pointed out in 1911 that electromagnetic radiation would cause electrons in the Rutherford's atom to fall upon the nucleus in no time. But he was also the one who "solved" the problem by suggesting... to ignore classical electrodynamics for quantized stationary orbits in his 1913 model. It was Rutherford's turn to return the favor by asking:"How does an electron decide what frequency it is going to vibrate at when it passes from one stationary state to another? It seems to me that you would have to assume that the electron knows beforehand where it is going to stop", see From Bohr's Atom to Electron Waves. Bohr's mathematician brother Harald intimated that young physicists at Göttingen considered Bohr's model to be too "bold" and "fantastic", and even Bohr himself admitted that it made no physical sense.
Things got worse in 1914, when Fowler made precision measurements on enhanced spectral lines of helium. They allowed him to determine that the "Rydberg constant" for helium was $4.0016$, not $4$, as Bohr's theory predicted. But what a a difference a small discrepancy can make:
"Bohr's response to this news was conclusive. He pointed out that his analysis of the circular motion had neglected the finite mass of the nucleus. He should really have taken the electron to have an effective mass equal to mM/(m + M). It is easy to verify that if this correction is made for both hydrogen and helium, the ratio changes from 4 to 4.0016!
This discovery won over most physicists to the point of view that Bohr was definitely on to something. Sir James Jeans remarked that the only justification for Bohr's postulates was "the very weighty one of success". According to Rosenfeld (Introduction to Niels Bohr Collected Works): "At Göttingen, that high place of mathematics and physics, where the sense for propriety was strong, the prevailing impression was one of scandal, or at least bewilderment, before the undeserved success of such high-handed disregard of the canons of formal logic." But all Germans didn't feel that way. Bohr's friend George Hevesy told Einstein about Bohr's identification of the Pickering-Fowler lines with helium. Einstein said: "This is an enormous achievement. The theory of Bohr must then be right".