The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $Ntoinfty$, we determine the exponent $ u$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $beta$ characterizing the growth of the amplitude of the order-parameter, as well as the initial-slip exponent $theta$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $ u, beta, theta$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period-doubled, are also found to oscillate in space at the wave-vector $omega/(2 v)$, $v$ being the velocity of the quasiparticles, and $omega$ being the drive frequency.
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