1902 Lord Kelvin (William Thomson) and J.J.Thomson independently suggested that positive charge in an atom was uniformly distributed over a sphere in which electrons were embedded. Thomson believed that the positive charge was associated with the bulk of the atom's mass and the negative electrons were in rings embedded in the positive charge. Read Kelvin's own words in this 1902 article.
1909....Geiger and Marsden's performed experiments bombarding gold foil with alpha particles. Rutherford's ideas connected with these experiments around 1906 - 1911 can be read in this paper.
1911 Rutherford developed a nuclear model of the atom to explain results of scattering experiments. Read about Rutherford's 1911 paper.
1911 Charles Glover Barkla found that X-rays were scattered in the same way as alpha-particles, and the results were used to calculate the number of electrons in atoms of light elements. He noted that, for the light elements, the number of electrons was about half the atomic weight. He was awarded the Nobel prize for Physics in 1917.
1913 Van den Broek expressed the realisation that the number of the position of an element in the periodic table was about half the atomic weight of the element. His article in Nature is referred to in a subsequent article by Soddy also in Nature.
1913 Geiger and Marsden used scattering experiment results to calculate the nuclear charge for gold and found a value close to what Rutherford had predicted on the basis of their earlier results. Rutherford reviewed this work in a 1920 lecture.
1913 Bohr first published his postulates concerning energy states of an electron in an atom, thus providing a model of a nuclear atom which overcame theoretical objections based on electromagnetic theory.
1913 Moseley made a systematic study of X-ray lines produced by 38 different elements.
1914 Kossel used Moseley's data to form a consistent picture of atomic structure.
1916 Franck and Hertz designed experiments which enabled them to observe electron energy levels and confirmed the Bohr theory that electrons in an atom can exist only in certain discrete energy states. Read their Nobel prize information.
1920 Chadwick's improved version of scattering experiments led to values of nuclear charge which were very close to current atomic number values for those elements. Chadwick was also responsible for discovering the neutron.
By this stage the numbers of electrons in atoms could be found, and the atomic number of elements (positive charge on the nucleus) provided an understanding of the periodic table.