  We study the early-time dynamics of a degenerate Bose gas after a sudden
quench of the interaction strength, starting from a weakly interacting gas. By
making use of a time-dependent generalization of the Nozi\`eres-Saint-James
variational formalism, we describe the crossover of the early-time dynamics
from shallow to deep interaction quenches. We analyze the coherent oscillations
that characterize both the density of excited states and the Tan's contact as a
function of the final scattering length. For shallow quenches, the oscillatory
behaviour is negligible and the dynamics is universally governed by the healing
length and the mean-field interaction energy. By increasing the final
scattering length to intermediate values, we reveal a universal regime where
the period of the coherent atom-molecule oscillations is set by the molecule
binding energy. For the largest scattering lengths we can numerically simulate
in the unitary regime, we find a universal scaling behaviour of the typical
growth time of the momentum distribution in agreement with recent experimental
observations [C. Eigen et al., Nature 563, 221 (2018)].
