Deep neural networks (DNNs) based automatic speech recognition (ASR) systems are often designed using expert knowledge and empirical evaluation. In this paper, a range of neural architecture search (NAS) techniques are used to automatically learn two types of hyper-parameters of state-of-the-art factored time delay neural networks (TDNNs): i) the left and right splicing context offsets; and ii) the dimensionality of the bottleneck linear projection at each hidden layer. These include the DARTS method integrating architecture selection with lattice-free MMI (LF-MMI) TDNN training; Gumbel-Softmax and pipelined DARTS reducing the confusion over candidate architectures and improving the generalization of architecture selection; and Penalized DARTS incorporating resource constraints to adjust the trade-off between performance and system complexity. Parameter sharing among candidate architectures allows efficient search over up to $7^{28}$ different TDNN systems. Experiments conducted on the 300-hour Switchboard corpus suggest the auto-configured systems consistently outperform the baseline LF-MMI TDNN systems using manual network design or random architecture search after LHUC speaker adaptation and RNNLM rescoring. Absolute word error rate (WER) reductions up to 1.0\% and relative model size reduction of 28\% were obtained. Consistent performance improvements were also obtained on a UASpeech disordered speech recognition task using the proposed NAS approaches.