The success of multi-task learning (MTL) can largely be attributed to the shared representation of related tasks, allowing the models to better generalise. In deep learning, this is usually achieved by sharing a common neural network architecture and jointly training the weights. However, the joint training of weighting parameters on multiple related tasks may lead to performance degradation, known as negative transfer. To address this issue, this work proposes an evolutionary multi-tasking neural architecture search (EMT-NAS) algorithm to accelerate the search process by transferring architectural knowledge across multiple related tasks. In EMT-NAS, unlike the traditional MTL, the model for each task has a personalised network architecture and its own weights, thus offering the capability of effectively alleviating negative transfer. A fitness re-evaluation method is suggested to alleviate fluctuations in performance evaluations resulting from parameter sharing and the mini-batch gradient descent training method, thereby avoiding losing promising solutions during the search process. To rigorously verify the performance of EMT-NAS, the classification tasks used in the empirical assessments are derived from different datasets, including the CIFAR-10 and CIFAR-100, and four MedMNIST datasets. Extensive comparative experiments on different numbers of tasks demonstrate that EMT-NAS takes 8% and up to 40% on CIFAR and MedMNIST, respectively, less time to find competitive neural architectures than its single-task counterparts.