In this paper, the introduction of interference cancellation in full-duplex joint radar and communication receivers is analysed. More specifically, a focus is made on scenarios in which the receiver simultaneously receives radar echoes from the environment, and communication signals from other joint radar and communication transceivers. First, the phase-coded frequency modulated continuous wave waveform designed for integrated sensing and communication is presented. Then, simple structures in which interference cancellation is only applied at the radar or communication function are proposed, relying on the information gathered at the other function. The detection probability and bit error rate improvements are analysed numerically w.r.t. different system parameters, such as the communication constellation, or the number of transmitted pulses. The introduction of error correcting codes is also considered. Next, iterative interference cancellation structures are investigated. Thanks to multiple interference cancellation layers, both the radar and communication performance are improved, and the robustness of the system to any scenario is enhanced. This is shown numerically through the analysis of the performance achieved by simple and iterative structures, which are compared for different radar echo to communication signal power ratio. Finally, a dynamic automotive scenario is considered. Leveraging on previous radar measurements, the parameters of the radar echo are inferred, and it is reconstructed beforehand, enabling to remove the first radar processing block. The complexity of the iterative structures is thus reduced, at the price of a slight performance reduction. A dynamic automotive scenario is considered, highlighting the impact of the tracking of the next vehicle ahead on the system.