This question is not directly related to, but was inspired by, this question. We know that a finitely generated residually nilpotent group is residually of prime-power order. However, we may need to use different primes for different elements. Classes of groups for which residual nilpotence forces there to be a single prime that will do for all elements (i.e., for which the group in question must be residually $p$-finite, for some $p$) seem to be interesting, and include, for instance, free products of cyclic groups.

Is there a (non-cyclic) one-relator group that is residually nilpotent, but is not residually a finite $p$-group, for any prime number $p$?

Such a group must be torsion-free, with trivial centre.