Cooperative Intelligent Transport Systems (C-ITS) enable information to be shared wirelessly between vehicles and infrastructure in order to improve transport safety and efficiency. Delivering C-ITS services using existing cellular networks offers both financial and technological advantages, not least since these networks already offer many of the features needed by C-ITS, and since many vehicles on our roads are already connected to cellular networks. Still, C-ITS pose stringent requirements in terms of availability and latency on the underlying communication system; requirements that will be hard to meet for currently deployed 3G, LTE, and early-generation 5G systems. Through a series of experiments in the MONROE testbed (a cross-national, mobile broadband testbed), the present study demonstrates how cellular multi-access selection algorithms can provide close to 100% availability, and significantly reduce C-ITS transaction times. The study also proposes and evaluates a number of low-complexity, low-overhead single-access selection algorithms, and shows that it is possible to design such solutions so that they offer transaction times and availability levels that rival those of multi-access solutions.