We aim to determine the physical properties of OB stars from the multi-epoch Binarity at LOw Metallicity (BLOeM) spectroscopic survey of the Small Magellanic Cloud using the Very Large Telescope/Fibre Large Array Multi-Element Spectrograph. We apply a pipeline designed to analyse large spectroscopic samples of OB stars to the co-added, initial nine epochs of the BLOeM survey, utilizing grids of synthetic model spectra computed with the stellar atmosphere code FASTWIND. 69 OB stars are excluded from the analysis owing to disc emission or significant contamination by secondaries in SB2 binaries. We determine physical properties of 778 OB stars, including $T_{\rm eff}$, $\log g$, $\log L/{\mathrm{L}}_{\odot }$, and $\upsilon _{\rm e} \sin i$. There appears to be a bimodality in $\upsilon _{\rm e} \sin i$ of single O stars, while $\upsilon _{\rm e} \sin i$ distributions of OB stars are strikingly different for single (median 78 km s$^{-1}$) and binary (median 200 km s$^{-1}$) systems. Inferred temperatures are broadly in agreement with literature results for stars in common, plus results from a grid-based automization tool for a subset of O and early B stars, although uncertainties are larger for surface gravities. Rotational velocities are broadly in line with an independent tool applied to the same subset. We recover the anticipated lower mass cut-off at 8 $\mathrm{ M}_{\odot }$ from the survey design using a Bayesian inference method coupled with SMC metallicity evolutionary models, with median masses of 12.6 $\mathrm{ M}_{\odot }$ (19.8 $\mathrm{ M}_{\odot }$) for B-type (O-type) stars. Spectroscopic masses exceed evolutionary masses, albeit with large uncertainties in surface gravities. We also provide an updated catalogue of O stars in the SMC since half of the 159 BLOeM O stars are newly classified as O-type stars.