A problem related to the development of an algorithm designed to find an architecture of artificial neural network used for black-box modelling of dynamic systems with time delays has been addressed in this paper. The proposed algorithm is based on a well-known NeuroEvolution of Augmenting Topologies (NEAT) algorithm. The NEAT algorithm has been adjusted by allowing additional connections within an artificial neural network and developing original specialised evolutionary operators. This resulted in a compromise between the size of neural network and its accuracy in capturing the response of the mathematical model under which it has been learnt. The research involved an extended validation study based on data generated from a mathematical model of an exemplary system as well as the fast processes occurring in a pressurised water nuclear reactor. The obtaining simulation results demonstrate the high effectiveness of the devised neural (black-box) models of dynamic systems with time delays.