In the present work, we demonstrate that an anomalous weak ferromagnetism occurs in the composition range 0.50 <= x <= 0.62 in nanostructured thin films of Nd1-xSrxMnO3 (NSMO). Nanocrystalline thin films of overdoped manganite Nd1-xSrxMnO3 (x similar to 0.50, 0.55, 0.60, and 0.62) are grown on single-crystal LaAlO3 (001) substrates by using a nebulized chemical spray pyrolysis technique. These single phase films possess an average crystallite size similar to 15 nm, and the width of the grain boundaries is similar to 1 - 2 nm. In the composition range 0.50 <= x <= 0.62, the ground state of NSMO is an A-type antiferromagnetic (AFM) metallic state. However, in the present films, a broad paramagnetic (PM)-to-ferromagnetic (FM) transition is observed in all the compositions, at T-C similar to 226 K for x = 0.50 and at 235 K for rest of the samples. All the films show a well-defined M-H hysteresis loop at 5 K. The coercivity (H-C) of these films is much larger than those having compositions in the range 0.35 < x < 0.45. For x = 0.50 (0.62), the films H-C = -1210 (-1484) Oe and +1162 (+1476) Oe. The asymmetry in the coercivity suggests the presence of a weak exchange bias effect in these films. The FM ground state in these films is observed to have a smaller magnetic moment per Mn atom than the expected full moment from the rigid model; we term this as a weak ferromagnetic (WFM) state arising due to the destabilization of the AFM ordering. We propose a possible scenario based on the combined effect of spin reorganization and enhanced orbital disordering in nanosized manganites to explain the observed anomalous weak ferromagnetism in the A-type AFM spin ordered region.