We present the confirmation of a hot super-Neptune with an exterior Neptune companion orbiting a bright (V = 10.1 mag) F-dwarf identified by the Transiting Exoplanet Survey Satellite (TESS). The two planets, observed in sectors 45, 46, and 48 of the TESS extended mission, are $4.74_{-0.14}^{+0.16}$ and $3.86_{-0.16}^{+0.17}$ R⊕ with $5.4588385_{-0.0000072}^{+0.0000070}$ and $17.8999_{-0.0013}^{+0.0018}$ d orbital periods, respectively. We also obtained precise space-based photometric follow-up of the system with ESA's CHaracterising ExOplanets Satellite to constrain the radius and ephemeris of TOI-5126 b. TOI-5126 b is located in the 'hot Neptune Desert' and is an ideal candidate for follow-up transmission spectroscopy due to its high-predicted equilibrium temperature (Teq = ${1442}_{-40}^{+46}$ K) implying a cloud-free atmosphere. TOI-5126 c is a warm Neptune (Teq = $971_{-27}^{+31}$ K) also suitable for follow-up. Tentative transit timing variations have also been identified in analysis, suggesting the presence of at least one additional planet, however this signal may be caused by spot-crossing events, necessitating further precise photometric follow-up to confirm these signals.