The inner flow geometry in MAXI J1820+070 during hard and hard-intermediate states

[Abridged] Context: We present a systematic X-ray spectral-timing study of the recently discovered, exceptionally bright black hole X-ray binary system MAXI J1820+070. Our analysis focuses on the first part of the 2018 outburst, covering the rise throughout the hard state, the bright hard and hard-intermediate states, and the transition to the soft-intermediate state. Aims: We address the issue of constraining the geometry of the innermost accretion flow and its evolution throughout an outburst. Methods: We employed two independent X-ray spectral-timing methods applied to the NICER data of MAXI J1820+070. We first identified and tracked the evolution of a characteristic frequency of soft X-ray reverberation lags. Then, we studied the spectral evolution of the quasi-thermal component responsible for the observed thermal reverberation lags. Results: The frequency of thermal reverberation lags steadily increases throughout most of the outburst, implying that the relative distance between the X-ray source and the disc decreases as the source softens. However, near transition this evolution breaks, showing a sudden increase (decrease) of lag amplitude (frequency). The evolution of the quasi-thermal component in high-frequency covariance spectra is consistent with a steady decrease of the inner radius of the disc as the source softens. Conclusions: The behaviour of thermal reverberation lags near transition might be related to relativistic plasma ejections detected at radio wavelengths later in the outburst, possibly representing the precursor to such events. Throughout most of the hard and hard-intermediate states the disc is consistent with being truncated (with an inner radius $R_{\rm in}>\sim 10 R_{\rm g}$), reaching close to the innermost stable circular orbit only near transition.