  [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.
