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PSR J0514$-$4002A is a binary millisecond pulsar located in the globular cluster NGC 1851. The pulsar has a spin period of 4.99 ms, an orbital period of 18.8 days, and is in a very eccentric ($e = 0.89$) orbit around a massive companion. In this work, we present the updated timing analysis of this system, obtained with an additional 1 yr of monthly observations using the Giant Metrewave Radio Telescope and 2.5 yrs of observations using the MeerKAT telescope. This has allowed for a precise measurement of the proper motion of the system, implying a transverse velocity of $30\,\pm\,7\,\mathrm{km}\,\mathrm{s}^{-1}$ relative to the cluster. This is smaller than the cluster's escape velocity and consistent with the pulsar's association to NGC 1851. We have confirmed a large second spin frequency derivative and large associated jerk, which has increased the spin frequency derivative by a factor of 27 since the mid-2000s. The third spin frequency derivative showed that the strength of this jerk has increased by $\sim 65\%$ in the same time period. We take the effect of the changing acceleration into account and this allows for much improved estimates of the orbital period derivative. The large and fast-increasing jerk implies the presence of a third body in the vicinity of the pulsar (no counterpart is detectable within distance limit in HST images). Based on our measured parameters, we constrain the mass, distance and orbital parameters for this third body. The induced tidal contributions to the post-Keplerian parameters are small, and the precise measurement of these parameters allowed us to obtain precise mass measurements for the system: $M_\mathrm{tot} = 2.4734(3)$ M$_{\odot}$, $M_\mathrm{p} = 1.39(3)$ M$_{\odot}$, $M_\mathrm{c} = 1.08(3)$ M$_{\odot}$. This indicates that the pulsar's companion is a massive white dwarf and resolves the earlier ambiguity regarding its nature.