Reduced-order modelling and low-dimensional surrogate models generated using machine learning algorithms have been widely applied in high-dimensional dynamical systems to improve the algorithmic efficiency. In this paper, we develop a system which combines reduced-order surrogate models with a novel data assimilation (DA) technique used to incorporate real-time observations from different physical spaces. We make use of local smooth surrogate functions which link the space of encoded system variables and the one of current observations to perform variational DA with a low computational cost. The new system, named generalised latent assimilation can benefit both the efficiency provided by the reduced-order modelling and the accuracy of data assimilation. A theoretical analysis of the difference between surrogate and original assimilation cost function is also provided in this paper where an upper bound, depending on the size of the local training set, is given. The new approach is tested on a high-dimensional (CFD) application of a two-phase liquid flow with non-linear observation operators that current Latent Assimilation methods can not handle. Numerical results demonstrate that the proposed assimilation approach can significantly improve the reconstruction and prediction accuracy of the deep learning surrogate model which is nearly 1000 times faster than the CFD simulation.