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 Tans 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)].
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