Moving horizon estimation (MHE) solves a constrained dynamic optimisation problem. Including nonlinear dynamics into an optimal estimation problem generally comes at the cost of tackling a non-convex optimisation problem. Here, a particular model formulation is proposed in order to convexify a class of nonlinear MHE problems. It delivers a linear time-varying (LTV) model that is globally equivalent to the nonlinear dynamics in a noise-free environment, hence the optimisation problem becomes convex. On the other hand, in the presence of unknown disturbances, the accuracy of the LTV model degrades and this results in a less accurate solution. For this purpose, some assumptions are imposed and a homotopy-based approach is proposed in order to transform the problem from convex to non-convex, where the sequential implementation of this technique starts with solving the convexified MHE problem. Two simulation studies validate the efficiency and optimality of the proposed approach with unknown disturbances.