We report on the Mössbauer spectra and magnetization properties of single-crystalline ${\left({\text{BiFeO}}_3\right)}_x\text{−}{\left({\text{BaTiO}}_3\right)}_{1-x}$ solid solution nanostructures in the form of nanocubes, measuring approximately 150 to 200 nm on a side, prepared by a molten salt solid-state reaction method in the compositional range wherein $0.5\le x\le 1$. Powder x-ray diffraction (XRD) and monochromatic synchrotron XRD studies indicate products of high purity, which undergo gradual, well-controlled structural transformations from rhombohedral to tetragonal structures with decreasing “$x$.” For all solid solution products, room-temperature magnetization studies exhibit hysteretic behavior with remnant magnetization values of $M_{\text{r}}\ge 0.32\text{emu}/\text{g}$, indicating that the latent magnetization locked within the toroidal spin structure of ${\text{BiFeO}}_3$ has been released. Room-temperature Mössbauer spectra show composition-dependent characteristics with decreasing magnetic hyperfine field values and increasing absorption linewidths due to a decrease in the magnetic exchange interaction strength with decreasing $x$. For the lowest $x=0.5$ composition studied, the Mössbauer spectra show paramagnetic behavior, indicating a Néel temperature for this composition below 300 K. However, room-temperature magnetization studies with applied fields of up to 50 kOe show hysteretic behavior for all compositions, including the $x=0.5$ composition, presumably due to field-induced ordering. Furthermore, hysteresis loops for all compositions exhibit smaller coercivities at 10 K than at 300 K, an observation that may suggest the presence of magnetoelectric coupling in these systems.