Deep learning (DL) has achieved state-of-the-art results on benchmark datasets and been most popular and widely accepted for both industrial and research problems. Since the success of deep learning, particularly across multiple domains, and successful implementations by tech giants like Apple, Microsoft, Facebook, etc., the research focus is now diverted towards optimising DL process by using various conventional and unconventional approaches. One such approach is using evolutionary computation techniques to improve the training and learning of DL. This paper, for the first time, proposes a novel multi-population competitive and cooperative neuroevolution approach for evolving optimised deep neural networks (DNNs) to provide a warm start to the DL process. Two separate coevolutionary strategies are used on two different populations that are collectively working towards producing optimised DNNs. Furthermore, three new parameters are proposed to avoid premature convergence and to maintain genetic diversity which is considered to be one of the major problems for population-based approaches. The proposed approach is tested on four different and diversified datasets, i.e. IRIS, MNIST, prostate cancer gene expression and synthetic hierarchical dataset. The experiments are carried out with four different strategies (of two different sizes) using standard DNNs, DNNs evolved with coevolution strategies and the proposed multi-population competitive and cooperative evolutionary (evolutionary deep neural networks or simply EDN) approach. The proposed EDN approach dominated the results by reducing the training and execution times while maintaining the best classification accuracy among all other approaches for all four datasets. The classification accuracy with the proposed approach is 98.3%, 98.7%, 82.2% and 89.1% for IRIS, MNIST, gene expression dataset and synthetic dataset, respectively. With the proposed EDN, the execution time for training and testing (including evolving DNNs) is considerably reduced by over 50% for MNIST and synthetic dataset, whereas it is 15% for IRIS and gene expression datasets when compared to standard deep neural networks with random weights and fixed topology.