We report $^{139}$La nuclear magnetic resonance studies performed on a La$_{1.875}$Ba$_{0.125}$CuO$_4$ single crystal. The data show that the structural phase transitions (high-temperature tetragonal $rightarrow$ low-temperature orthorhombic $rightarrow$ low-temperature tetragonal phase) are of the displacive type in this material. The $^{139}$La spin-lattice relaxation rate $T_1^{-1}$ sharply upturns at the charge-ordering temperature $T_text{CO}$ = 54 K, indicating that charge order triggers the slowing down of spin fluctuations. Detailed temperature and field dependencies of the $T_1^{-1}$ below the spin-ordering temperature $T_text{SO}$ = 40 K reveal the development of enhanced spin fluctuations in the spin-ordered state for $H parallel [001]$, which are completely suppressed for large fields along the CuO$_2$ planes. Our results shed light on the unusual spin fluctuations in the charge and spin stripe ordered lanthanum cuprates.
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