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The idea of a spin angular momentum was first proposed in a 1925 publication by George Uhlenbeck and Samuel Goudsmit to explain hyperfine splitting in atomic spectra.
At what time was it measured that the subatomic particles have magnetic dipole moments?
Subatomic particles, other than electron, photon and proton were not yet discovered in 1925. On the proton (which at the time was only known as hydrogen nucleus "inside" other nuclei), Compton, who conjectured the electron spin in The magnetic electron (1921), wrote:
"Besides the molecule and the atom we have the other two fundamental divisions of matter, the atomic nucleus and the electron. The sign of the Richardson-Barnett effect indicates that it is negative electricity which is chiefly responsible for magnetic effects, which makes the view that the positive nucleus is the elementary magnet difficult to defend."
Uhlenbeck and Goudsmit followed Compton, even their follow-up expose in Nature Spinning Electrons and the Structure of Spectra only talks of spinning electrons. As for the photon, according to Experimental proof of the spin of the photon by Raman-Bhagavantam, Bose considered the possibility back in 1924
"In his well-known derivation of the Planck radiation formula from quantum statistics, Prof. Bose obtained an expression for the number of cells in phasespace occupied by the radiation, and found himself obliged to multiply it by a numerical factor $2$ in order to derive from it the correct number of possible arrangements of the quantum in unit volume. The paper as published did not contain a detailed discussion of the necessity for the introduction of this factor, but we understand from a personal communication by Prof. Bose that he envisaged the possibility of the quantum possessing besides energy $h\nu$ and linear momentum $h\nu/c$ also an intrinsic spin or angular momentum $\pm h/2\pi$ round an axis parallel to the direction of its motion. The weight factor $2$ thus arises from the possibility of the spin of the quantum being either right-handed or left-handed, corresponding to the two alternative signs of the angular momentum."
For further developments see Milner, A Short History of Spin:
"In the 1930s, Rabi and collaborators (inc. N. Ramsey and J. Za charias) using molecular beams in a weak magnetic field measured the magnetic moments and nuclear spins of hydrogen, deuterium, and heavier nuclei [22]. By the end of the 1940s, the nuclear shell model had been established [23]. This explained the properties and structure of atomic nuclei and underscored the essential role of proton and neutron spin. A key aspect was the strong role of spin-orbit coupling, which was suggested to Goeppert-Mayer by a question from Fermi... By the middle of the twentieth century, the intrinsic spin of subatomic particles was a cornerstone of the physicist’s theoretical understanding of the fundamental structure of matter."
