The predictive power of Convolutional Neural Networks (CNNs) has been an integral factor for emerging latency-sensitive applications, such as autonomous drones and vehicles. Such systems employ multiple CNNs, each one trained for a particular task. The efficient mapping of multiple CNNs on a single FPGA device is a challenging task as the allocation of compute resources and external memory bandwidth needs to be optimised at design time. This paper proposes f-CNNx, an automated toolflow for the optimised mapping of multiple CNNs on FPGAs, comprising a novel multi-CNN hardware architecture together with an automated design space exploration method that considers the user-specified performance requirements for each model to allocate compute resources and generate a synthesisable accelerator. Moreover, f-CNNx employs a novel scheduling algorithm that alleviates the limitations of the memory bandwidth contention between CNNs and sustains the high utilisation of the architecture. Experimental evaluation shows that f-CNNx's designs outperform contention-unaware FPGA mappings by up to 50% and deliver up to 6.8x higher performance-per-Watt over highly optimised GPU designs for multi-CNN systems.