Certain residual properties of bounded nilpotent groups and their tree products

Let $\mathfrak{P}$ be a~non-empty set of~primes. We prove that any $\mathfrak{P}$\nobreakdash-bounded nilpotent group is~$\mathfrak{P}$\nobreakdash-potent, and~the~tree product~$T$ of~a~finite number of~$\mathfrak{P}$\nobreakdash-bounded nilpotent groups with~proper locally cyclic edge subgroups is~residually a~finite $\mathfrak{P}$\nobreakdash-group if and~only if any vertex group of~$T$ has no~$\mathfrak{P}^{\prime}$\nobreakdash-torsion and~any edge subgroup of~$T$ is~$\mathfrak{P}^{\prime}$\nobreakdash-isolated in~the~vertex group containing~it. We prove also that the~tree product of~a~finite number of~groups with~locally cyclic edge subgroups is~residually a~finite $p$\nobreakdash-group if~all its vertex groups have this property and~any edge subgroup is~separable in~the~corresponding vertex group by~the~class of~finite $p$\nobreakdash-groups.