A search is conducted for a new Beyond-the-Standard-Model dark vector boson or a new light pseudoscalar boson using events where a Higgs boson with mass 125 GeV decays to four leptons ($\ell =$ $e$ or $\mu$). This decay is presumed to occur via an intermediate state which contains one or two on-shell, promptly decaying bosons: $H\rightarrow ZZ_{d} \rightarrow 4\ell$ or $H\rightarrow Z_dZ_d \rightarrow 4\ell$, or via two light pseudoscalar bosons : $H\rightarrow aa \rightarrow 4\mu$. The search uses $pp$ collision data collected with the ATLAS detector with an integrated luminosity of 36.1~fb$^{-1}$ at the centre-of-mass energy $\sqrt{s}=13$ TeV. Dedicated selections are defined for each of the three decay channels. For the first case, a total of 102 events are observed with $87\pm 7$ predicted by the Standard Model in a search region sensitive to bosons with a mass between 15 GeV~and 55 GeV~across the $2\ell2\mu$ and $2\ell 2e$ decay channels. For the second case, which is performed in a search region sensitive to bosons with a mass between 15 GeV~and 60 GeV, a total of 6 events are observed across the $4e$, $2e2\mu$ and $4\mu$ decay channels, with a Standard Model prediction of $3.9 \pm 0.3$. The search is extended to boson masses between 1 GeV and 15 GeV, excluding masses close to known dilepton mass resonances, for $4\mu$ final states only. In this extended search region, no events are observed with a prediction of $0.4\pm0.1$. Upper limits at 95% confidence level are set on model-independent fiducial cross sections, and on the Higgs boson decay branching ratios to vector and pseudoscalar bosons in two benchmark models.

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