Swarms of autonomous devices are increasing in ubiquity and size, making the need for rethinking their hardware-software system stack critical. We present HiveMind, the first swarm coordination platform that enables programmable execution of complex task workflows between cloud and edge resources in a performant and scalable manner. HiveMind is a software-hardware platform that includes a domain-specific language to simplify programmability of cloud-edge applications, a program synthesis tool to automatically explore task placement strategies, a centralized controller that leverages serverless computing to elastically scale cloud resources, and a reconfigurable hardware acceleration fabric for network and remote memory accesses. We design and build the full end-to-end HiveMind system on two real edge swarms comprised of drones and robotic cars. We quantify the opportunities and challenges serverless introduces to edge applications, as well as the trade-offs between centralized and distributed coordination. We show that HiveMind achieves significantly better performance predictability and battery efficiency compared to existing centralized and decentralized platforms, while also incurring lower network traffic. Using both real systems and a validated simulator we show that HiveMind can scale to thousands of edge devices without sacrificing performance or efficiency, demonstrating that centralized platforms can be both scalable and performant.