Processive proteases, such as ClpXP in E. coli, are conserved enzyme assemblies that can recognize and rapidly degrade proteins. These proteases are used for a number of purposes, including degrading mistranslated proteins and controlling cellular stress response. However, proteolytic machinery within the cell is limited in capacity and can lead to a bottleneck in protein degradation, whereby many proteins compete ('queue') for proteolytic resources. Previous work has demonstrated that such queueing can lead to pronounced statistical relationships between different protein counts when proteins compete for a single common protease. However, real cells contain many different proteases, e.g. ClpXP, ClpAP, and Lon in E. coli, and it is not clear how competition between proteins for multiple classes of protease would influence the dynamics of cellular networks. In the present work, we theoretically demonstrate that a multi-protease proteolytic bottleneck can substantially couple the dynamics for both simple and complex (oscillatory) networks, even between substrates with substantially different affinities for protease. For these networks, queueing often leads to strong positive correlations between protein counts, and these correlations are strongest near the queueing theoretic point of balance. Furthermore, we find that the qualitative behavior of these networks depends on the relative size of the absolute affinity of substrate to protease compared to the cross affinity of substrate to protease, leading in certain regimes to priority queue statistics.