The young TOI-451 planetary system, aged 125 Myr, provides a unique opportunity to test theories of planetary internal structures and atmospheric mass-loss through examination of its three transiting planets. We present an exhaustive photometric and spectroscopic follow-up to determine the orbital and physical properties of the system. We perform multidimensional Gaussian Process regression with the code pyaneti on spectroscopic time-series and NGTS/LCO light curves to disentangle the stellar and planetary signal in ESPRESSO radial velocities. We show how contemporaneous photometry serves as an activity indicator to inform RV modelling within a multidimensional Gaussian Processes framework. We argue that this can be exploited when spectroscopic observations are adversely affected by low signal-to-noise and/or poor sampling. We estimate the Doppler semi-amplitudes of $k_{\rm b}=$ $2.6_{-1.2}^{+1.1}$ $\mathrm{m\, s^{-1}}$, $k_{\rm c}=$ $1.2_{-0.8}^{+1.0}$ $\mathrm{m\, s^{-1}}$ , and $k_{\rm d}=$ $2.7 \pm 1.2$ $\mathrm{m\, s^{-1}}$ . This translates in 2$\sigma$ mass estimates for TOI-451 b and d of $M_{\rm b}=$ $4.7_{-2.2}^{+2.1}$ $\mathrm{M_{\rm{\oplus }}}$ and $M_{\rm d}=$ $10.2_{-4.5}^{+4.6}$ $\mathrm{M_{\rm{\oplus }}}$; as well as a mass upper limit for TOI-451 c of $M_{\rm c} \lt 11.5\, M_{\rm{\oplus }}$. The derived planetary properties suggest that planets c and d contain significant hydrogen-rich envelopes. The inferred parameters of TOI-451 b are consistent with either a rocky world that still retains a small hydrogen envelope or a water world. These insights make the TOI-451 system an ideal laboratory for future follow-up studies aimed at measuring atmospheric compositions, detecting atmospheric mass-loss signatures, and further exploring planetary formation and evolution processes.