We investigate on the same footing the time-dependent electronic transport properties and vibrational dynamics of a molecular junction. We show that fluctuations of both the molecular vibron displacement and the electronic current across the junction undergo damped oscillations towards the steady-state. We assign the former to the onset of electron tunneling events assisted by vibron-emission. The time-dependent build-up of electron-hole correlations is revealed as a departure of the charge-transfer statistics from the generalized-binomial one after a critical time tc. The phonon-back action on the tunneling electrons is shown to amplify and accelerate this build-up mechanism.