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.
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