  [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.
