Historically, the discovery of naturally-occurring isotopes of chemical elements by means of the mass-spectrograph provided a correct explanation for the fractional character of the experimentally-measured values for the atomic weights of many chemical elements -- as compared with values that might have been expected if the atoms of a given element had been all alike as integer multiples of the atomic weight of hydrogen, or (later) as integers on a scale obtained by defining the atomic weight of oxygen O = 16.
Thus, Francis W Aston (1877-1945) discovered, by means of his mass-spectrograph (1919), many isotopes -- a designation coined with a Greek etymology for 'equal place', and signifying chemical elements occupying the same place in the periodic table but with atoms that differ in atomic weight: Aston was later awarded a Nobel Prize for his discoveries.
Aston's book 'Isotopes' (1922) (https://archive.org/details/isotopes00asto) gives a short history of the measurement of atomic weights and of attempted explanations of these weights, starting with the hypotheses of Dalton (1803) and of Prout (about ten years after). Dalton's far-reaching atomic theory or hypothesis included the idea that all atoms of a given element were alike: and Prout's hypothesis had been that all atoms are aggregations of atoms of hydrogen.
Later on it became clear, with increasing accuracy of measurement, that many atomic weights could not possibly represent whole-number aggregates of hydrogen, and Dalton's hypothesis (so far as it related to the weights of atoms) and Prout's could not both be true. The weight of chemical opinion at that point, Aston tells us, was to give up Prout's idea and to prefer Dalton's. In retrospect, and with awareness of the hindsight of discovery in respect of isotopes, Aston labeled that decision as "as wise in principle as it was wrong in fact".
From Aston (1922): "The idea that atoms of the same element are are identical in weight could not be challenged by chemical methods, for the atoms are by definition chemically identical and numerical ratios were only to be obtained in such methods by the use of quantities of the element containing countless myriads of atoms. At the same time it is rather surprising, when we consider the complete absence of positive evidence in its support that no theoretical doubts were publicly expressed until late in the nineteenth century, first by Schutzenberger and then by Crookes, and that these doubts have been regarded, even up to the last few years, as speculative in the highest degree."
There was a further attempt at explaining among other phenomena the diversity of atomic weights, William Crookes' idea (1886) of meta-elements. Crookes reached the idea that chemically inseparable elements might consist of a variety of atoms. This idea seemed at first to apply to the results of Crookes' study of yttria (oxide of yttrium) -- "But as more and more refined chemical methods were applied, the rare earths one after another yielded to analysis and the different spectra observed by Crookes were shown to be due to the fact that he was dealing with a mixture of real elements, each of which had a characteristic spectrum and a definite atomic weight. The theory of meta-elements was therefore abandoned and the problem of fractional atomic weight remained unsolved."
Accordingly, it seems that there was essentially no viable explanation of the well-known fractional atomic weights until the mass-spectrographic discovery of isotopes.
Thus, the fact that many standard atomic weights are not close to integers on a scale where hydrogen=1, or any closely similar scale, was explained (satisfactorily), before knowledge of the neutron, by the discovery that isotopes of the same chemical element actually exist with different atomic weights and a variety of natural relative abundances. It can also be seen from Aston's work that the neutron had essentially no role in this discovery. For example, the existence of chlorine-35 and chlorine-37 with about 3:1 relative natural abundance was shown by the mass-spectrographic work, and it explains the accepted value of about 35.5 for the atomic weight of chlorine.
(There is naturally an onward question how such isotopes could exist, but that matter is separate from the question answered by experimental demonstration that such isotopes exist in fact. The onward question 'how' involves among other things the distinction between atomic weight and the atomic numbers discovered experimentally in 1913 by H G Moseley, as well as the evolution of theories about atomic structure, theories that came to include, of course, the neutron.)