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Context. B hypergiants (BHGs) are important for understanding high-mass stellar evolution. While they are in a similar parameter space of B supergiants (BSGs), some BHGs are known to be luminous blue variables (LBVs). Their spectra with absorption and emission features resemble Of/WNh stars. Yet, their wind physics and their evolutionary connections are not clear. Aims. In this study, we aim to understand (i) the atmospheric and wind structure, (ii) the wind-launching and -driving mechanisms, and (iii) the spectrum formation of early-type BHGs. As an observational prototype, we use zet1 Sco (B1.5Ia+). Methods. Using the atmosphere code PoWRhd, we calculated the first hydrodynamically consistent models at the BHG domain. They give insights into the radiative driving of the calculated wind regimes and enable us to study the influence of clumping and X-rays on the resulting wind structure. Results. Our consistent model reproduces the main spectral features of zet1 Sco. The obtained mass-loss rate is higher than that of similar spectral type BSGs. However, the wind optical depth of BHGs is way below unity, making them less of a transition type. To reproduce zet1 Sco's spectrum, we needed low clumping with subsonic onset. The wind has a shallow-gradient velocity profile, deviating from the beta law, and is mainly driven by Fe III opacity. Conclusions. Our study suggests that despite more mass loss, early-type Galactic BHGs have winds relatively similar to BSGs. Their winds are not thick enough to characterize them as "transition-type" stars, unlike Of/WNh, implying that emission features arise more easily in cooler than in hotter stars. The spectral BHG appearance is likely connected to atmospheric inhomogeneities below the sonic point. To reach an appearance similar to LBVs, BHGs need to be either closer to the Eddington limit or have higher wind clumping than inferred for zeta1 Sco.