  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.
