Structural properties and water dissolution of six sodium–aluminum–phosphate (NAP) glasses have been investigated before and after irradiation by a gamma-ray source based on 60Co. Two of these samples were of simple composition, and four samples had a complex composition with radionuclide simulants representing actinides, fission, and activated corrosion products. Samples of the simple composition are fully vitreous, whereas samples of the complex composition contained up to 10 vol.% of aluminum–phosphate, AlPO4, and traces of ruthenium dioxide, RuO2. Based on the study of pristine and irradiated glasses, it was established that the radiation dose of 62 million Gray had practically no effect on the phase composition and structure of samples. At the same time, the rate of leaching of elements from the irradiated samples by water was decreased by about two times.