The past decade has seen a dramatic increase in the degree, quality, and sophistication of control over quantum-mechanical interactions available between artificial degrees of freedom in a variety of experimental platforms. The geometrical structure of these interactions, however, remains largely constrained by the underlying rectilinear geometry of three-dimensional Euclidean space. At the same time, there has been growing interest in exploring many-body dynamics in systems, such as the SYK model and tensor network models, for which the interaction structure bears little or no resemblance to Euclidean space. Inspired by these complementary developments, here we study a tunable, nonlocal spin network that can be engineered using cold atoms coupled to an optical cavity. The network exhibits two distinct notions of emergent geometry -- linear and treelike -- that can be accessed using a single tunable parameter. In either of these two extreme limits, we find a succinct description of the resulting dynamics in terms of two distinct metrics on the network, encoding a notion of either linear or treelike distance between spins. Moreover, at the crossover between these two regimes, the spin network becomes highly connected and exhibits signatures of fast scrambling. These observations highlight the essential role played by the geometry of the interaction structure in determining a system's dynamics, and raise prospects for novel studies of nonlocal and highly chaotic quantum dynamics in near-term experiments.