Ultracompact minihaloes (UCMHs) have been proposed as a type of dark matter substructure seeded by large-amplitude primordial perturbations and topological defects. UCMHs are expected to survive to the present era, allowing constraints to be placed on their cosmic abundance using observations within our own Galaxy. Constraints on their number density can be linked to conditions in the early Universe that impact structure formation, such as increased primordial power on small scales, generic weak non-Gaussianity, and the presence of cosmic strings. We use new constraints on the abundance of UCMHs from pulsar timing to place generalized limits on the parameters of each of these cosmological scenarios. At some scales, the limits are the strongest to date, exceeding those from dark matter annihilation. Our new limits have the added advantage of being independent of the particle nature of dark matter, as they are based only on gravitational effects.