We study how zealotry and nonlinear social impact affect consensus formation in the nonlinear voter model, evolutionary games, and the partisan voter model. In all three models, consensus is an absorbing state in finite populations, while co-existence is a possible outcome of the deterministic dynamics. We show that sufficiently strong zealotry -- i.e., the presence of agents who never change state -- can drive infinite populations to consensus in all three models. However, while evolutionary games and the partisan voter model permit zealotry-induced consensus for all values of their model parameters, the nonlinear voter model does not. Central to this difference is the shape of the social impact function, which quantifies how the influence of a group scales with size, and is therefore a measure of majority and minority effects. We derive general conditions relating the slope of this function at small group sizes to the local stability of consensus. Sublinear impact favours minorities and can override zealotry to prevent consensus, whereas superlinear impact promotes majorities and therefore facilitates consensus. We extend the analysis to finite populations, exploring the time-to-consensus, and the shape of quasi-stationary distributions.