  We use single crystal $^{63}$Cu NMR techniques to revisit the early $^{63}$Cu
NQR signature of charge order observed for La$_{1.875}$Ba$_{0.125}$CuO$_{4}$
($T_{\text{c}} =4$~K) [A. W. Hunt et al., Phys. Rev. Lett. {\bf 82}, 4300
(1999)]. We show that the growth of spin correlations is accelerated below
$\sim 80$~K, where the inverse Laplace transform (ILT) T$_{1}$ analysis of the
$^{139}$La NMR spin-lattice relaxation curve recently uncovered emergence of
the slow components in the lattice and/or charge fluctuations [P. M. Singer et
al., {\bf 101}, 174508 (2020)]. From the accurate measurements of the $^{63}$Cu
NMR signal intensity, spin echo decay $M(2\tau)$, spin-lattice relaxation rate
$^{63}1/T_1$, and its density distribution function $P(^{63}1/T_{1})$, we also
demonstrate that charge order at $T_{\text{charge}}\simeq 54$~K turns on strong
enhancement of spin fluctuations {\it within charge ordered domains}, thereby
making the CuO$_2$ planes extremely inhomogeneous. The charge ordered domains
grow quickly below $T_{\text{charge}}$, and the volume fraction $F_{\text{CA}}$
of the canonical domains unaffected by charge order gradually diminishes by
$\sim 35$~K. This finding agrees with our independent estimations of
$F_{\text{CA}}$ based entirely on the $^{139}$La ILTT$_{1}$ analyses, but is in
a stark contrast with much slower growth of charge ordered domains observed for
La$_{1.885}$Sr$_{0.115}$CuO$_{4}$ from its $T_{\text{charge}}\simeq 80$~K to
$T_{\text{c}}\simeq 30$~K.
