  We report $^{139}$La and $^{63}$Cu NMR investigation of the successive charge
order, spin order, and superconducting transitions in super-oxygenated
La$_2$CuO$_{4+y}$ single crystal with stage-4 excess oxygen order at
$T_{stage}\simeq 290$ K. We show that the stage-4 order induces tilting of
CuO$_6$ octahedra below $T_{stage}$, which in turn causes $^{139}$La NMR line
broadening. The structural distortion continues to develop far below
$T_{stage}$, and completes at $T_{charge}\simeq 60$ K, where charge order sets
in. This sequence is reminiscent of the the charge order transition in Nd
co-doped La$_{1.88}$Sr$_{0.12}$CuO$_4$ that sets in once the low temperature
tetragonal (LTT) phase is established. We also show that the paramagnetic
$^{63}$Cu NMR signals are progressively wiped out below $T_{charge}$ due to
enhanced low frequency spin fluctuations, but the residual $^{63}$Cu NMR
signals continue to exhibit the characteristics expected for optimally doped
superconducting CuO$_2$ planes. This indicates that charge order in
La$_2$CuO$_{4+y}$ does not take place uniformly in space. Low frequency Cu spin
fluctuations as probed by $^{139}$La nuclear spin-lattice relaxation rate are
mildly glassy, and do not exhibit critical divergence at $T_{spin}$($\simeq
T_{c}$)=42 K. These findings, including the spatially inhomogeneous nature of
the charge ordered state, are qualitatively similar to the case of
La$_{1.885}$Sr$_{0.115}$CuO$_4$ [T. Imai et al., Phys. Rev. B 96 (2017) 224508,
and A. Arsenault et al., Phys. Rev. B 97 (2018) 064511], but both charge and
spin order take place more sharply in the present case.
