This paper investigates the decoding process of asynchronous convolutional-coded physical-layer network coding (PNC) systems. Specifically, we put forth a layered decoding framework for convolutional-coded PNC consisting of three layers: symbol realignment layer, codeword realignment layer, and joint channel-decoding network coding (Jt-CNC) decoding layer. Our framework can deal with phase asynchrony and symbol arrival-time asynchrony between the signals simultaneously transmitted by multiple sources. A salient feature of this framework is that it can handle both fractional and integral symbol offsets; previously proposed PNC decoding algorithms (e.g., XOR-CD and reduced-state Viterbi algorithms) can only deal with fractional symbol offset. Moreover, the Jt-CNC algorithm, based on belief propagation (BP), is BER-optimal for synchronous PNC and near optimal for asynchronous PNC. Extending beyond convolutional codes, we further generalize the Jt-CNC decoding algorithm for all cyclic codes. Our simulation shows that Jt-CNC outperforms the previously proposed XOR-CD algorithm and reduced-state Viterbi algorithm by 2dB for synchronous PNC. For phase-asynchronous PNC, Jt-CNC is 4dB better than the other two algorithms. Importantly, for real wireless environment testing, we have also implemented our decoding algorithm in a PNC system built on the USRP software radio platform. Our experiment shows that the proposed Jt-CNC decoder works well in practice.