On functors preserving monoid objects

Here is a simple counterexample: take $C=D=\mathrm{Set}$, and let $\tilde{G}$ be the abelianization of a monoid. Then $\tilde{G}$ cannot arise from any $G:\mathrm{Set}\to\mathrm{Set}$, because the cardinality of the underlying set of $\tilde{G}(M)$ depends on the monoid structure of $M$, not just on its cardinality. For instance, take an abelian monoid and a nonabelian monoid of the same cardinality, say the symmetric group $S_3$ and the cyclic group $C_6$: then $U(S_3) \cong U(C_6)$, but $U \tilde{G}(S_3) \not\cong U\tilde{G}(C_6)$ since the latter still has 6 elements but the former has fewer; thus $U \tilde{G} \not\cong G U$ for any functor $G$.