We apply the reparameterisation trick to obtain a variational formulation of Bayesian inference in nonlinear ODE models. By invoking the linear noise approximation we also extend this variational formulation to a stochastic kinetic model. Our proposed inference method does not depend on any emulation of the ODE solution and only requires the extension of automatic differentiation to an ODE. We achieve this through a novel and holistic approach that uses both forward and adjoint sensitivity analysis techniques. Consequently, this approach can cater to both small and large ODE models efficiently. Upon benchmarking on some widely used mechanistic models, the proposed inference method produced a reliable approximation to the posterior distribution, with a significant reduction in execution time, in comparison to MCMC.