This paper proposes a video caching and transcoding strategy for delay-constrained content delivery in multi-tier wireless networks. Particularly for multimedia services whose content can be encoded into multiple quality versions and consists of multiple chunks, we present an approach of caching chunks of an identical file separately in different network layers with different qualities. Given this caching policy, a complete file can be retrieved by receiving its chunks from different network tiers; therefore, a video transcoding and delivery method is jointly designed with caching to maximize the expected content quality with a strict delay constraint. Motivated by that caching is proactively conducted in advance of many user requests, the joint optimization problem is decomposed into caching and transcoding subproblems. The optimal transcoding policy and its transcoded qualities at different network tiers are analytically found by linear programming for a given cache state. Using the closed-form transcoded qualities, the caching policy is optimized to maximize the expected content quality with respect to randomness of user locations and channel gains. Due to the difficulty of expectation, the caching policy of each network layer is independently obtained by water-filling first, and an alternating optimization method is used to find the joint optimal caching policy satisfying the cache size constraints. Simulation results show that the proposed scheme succeeds in delivering content of 5% to 10% higher expected quality to users than comparison schemes which do not use partitioning for caching. Simultaneously, our approach meets the delay requirement and effectively manages the trade-offs arising from the varying storage and computational capacities within hierarchies, as well as video diversity.