  We consider the time evolution of the renormalized Nelson model, which
describes $N$ bosons linearly coupled to a quantized scalar field, in the
mean-field limit of many particles $N\gg 1$ with coupling constant proportional
to $N^{-1/2}$. First, we show that initial states exhibiting Bose-Einstein
condensation for the particles and approximating a coherent state for the
quantum field retain their structure under the many-body time evolution.
Concretely, the dynamics of the reduced densities are approximated by solutions
of two coupled PDEs, the Schr\"odinger-Klein-Gordon equations. Second, we
construct a renormalized Bogoliubov evolution that describes the quantum
fluctuations around the Schr\"odinger-Klein-Gordon equations. This evolution is
used to extend the approximation of the evolved many-body state to the full
norm topology. In summary, we provide a comprehensive analysis of the Nelson
model that reveals the role of renormalization in the mean-field Bogoliubov
theory.
