The SARS-CoV-2 Delta (Pango lineage B.1.617.2) variant of concern spread globally, causing resurgences of COVID-19 worldwide. The emergence of the Delta variant in the UK occurred on the background of a heterogeneous landscape of immunity and relaxation of non-pharmaceutical interventions. Here we analyse 52,992 SARS-CoV-2 genomes from England together with 93,649 genomes from the rest of the world to reconstruct the emergence of Delta and quantify its introduction to and regional dissemination across England in the context of changing travel and social restrictions. Using analysis of human movement, contact tracing and virus genomic data, we find that the geographic focus of the expansion of Delta shifted from India to a more global pattern in early May 2021. In England, Delta lineages were introduced more than 1,000 times and spread nationally as non-pharmaceutical interventions were relaxed. We find that hotel quarantine for travellers reduced onward transmission from importations; however, the transmission chains that later dominated the Delta wave in England were seeded before travel restrictions were introduced. Increasing inter-regional travel within England drove the nationwide dissemination of Delta, with some cities receiving more than 2,000 observable lineage introductions from elsewhere. Subsequently, increased levels of local population mixing-and not the number of importations-were associated with the faster relative spread of Delta. The invasion dynamics of Delta depended on spatial heterogeneity in contact patterns, and our findings will inform optimal spatial interventions to reduce the transmission of current and future variants of concern, such as Omicron (Pango lineage B.1.1.529).