We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an ω-categorical algebra A. There are ω-categorical groups where this problem is undecidable. We show that if A is an ω-categorical semilattice or an abelian group, then the problem is in P or NP-hard. The hard cases are precisely those where Pol(A,≠) has a uniformly continuous minor-preserving map to the clone of projections on a two-element set. The results provide information about algebras A such that Pol(A,≠) does not satisfy this condition, and they are of independent interest in universal algebra. In our proofs we rely on the Barto-Pinsker theorem about the existence of pseudo-Siggers polymorphisms. To the best of our knowledge, this is the first time that the pseudo-Siggers identity has been used to prove a complexity dichotomy.