Aims: We consider cold self-similar magnetohydrodynamic (MHD) disc wind solutions to describe jets that are launched from the circumcompanion accretion discs in post-AGB binaries. Resulting predictions are matched to observations for five different post-AGB binaries. This both tests the physical validity of the MHD disc wind paradigm and reveals the accretion disc properties. Results: Many of the time-series' properties are reproduced well by the models, though systematic mismatches, such as overestimated rotation, remain. Four targets imply accretion discs that reach close to the secondary's stellar surface, while one is fitted with an unrealistically large inner radius of about 20 stellar radii. Some fits imply inner disc temperatures over 10 000 K, seemingly discrepant with a previous observational estimate from H band interferometry. This estimate is, however, shown to be biased. Fitted mass-accretion rates range from about 10^-6 to 10^-3 solar masses per year. Relative to the jets launched from young stellar objects (YSOs), all targets prefer winds with higher ejection efficiencies, lower magnetizations and thicker discs. Conclusions: Our models show that current cold MHD disc wind solutions can explain many of the jet-related Balmer alpha features seen in post-AGB binaries, though systematic discrepancies remain. This includes, but is not limited to, overestimated rotation and underestimated post-AGB circumbinary disc lifetimes. The consideration of thicker discs and the inclusion of irradiation from the post-AGB primary, leading to warm magnetothermal wind launching, might alleviate these.