[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 temperature of the quasi-thermal component in covariance spectra consistently increases throughout all the analysed observations. Conclusions: The behaviour of thermal reverberation lags near transition might be related to the relativistic plasma ejections detected at radio wavelengths, suggesting a causal connection between the two phenomena. 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.
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