Hyperfine-structure parameters and isotope shifts of At195-211 have been measured for the first time at CERN-ISOLDE, using the in-source resonance-ionization spectroscopy method. The hyperfine structures of isotopes were recorded using a triad of experimental techniques for monitoring the photo-ion current. The Multi-Reflection Time-of-Flight Mass Spectrometer, in connection with a high-resolution electron multiplier, was used as an ion-counting setup for isotopes that either were affected by strong isobaric contamination or possessed a long half-life; the ISOLDE Faraday cups were used for cases with high-intensity beams; and the Windmill decay station was used for short-lived, predominantly α-decaying nuclei. The electromagnetic moments and changes in the mean-square charge radii of the astatine nuclei have been extracted from the measured hyperfine-structure constants and isotope shifts. This was only made possible by dedicated state-of-the-art large-scale atomic computations of the electronic factors and the specific mass shift of atomic transitions in astatine that are needed for these extractions. By comparison with systematics, it was possible to assess the reliability of the results of these calculations and their ascribed uncertainties. A strong deviation in the ground-state mean-square charge radii of the lightest astatine isotopes, from the trend of the (spherical) lead isotopes, is interpreted as the result of an onset of deformation. This behavior bears a resemblance to the deviation observed in the isotonic polonium isotopes. Cases for shape coexistence have been identified in At197,199, for which a significant difference in the charge radii for ground (9/2-) and isomeric (1/2+) states has been observed.