If $G$ is a pronilpotent group, then $G$ is the product of its Sylow subgroups (like in the finite case). Suppose that $G$ is finitely generated, then can be a Sylow subgroups infinitely generated? Intuitively, I think not but I cannot prove (or disprove).
Actually, I think is enough to check this in the finite case: if $G$ is a finite nilpotent group $n$-generated, then is any Sylow subgroup at most $n$-generated?
I think yes. Writting $$G = \prod_p S_p,$$ then $$G/\prod_{q \neq p}S_q \simeq S_p$$ and $G/\prod_{q \neq p}S_q$ is at most $n$-generated.