To identify and characterize roles for SARS-CoV-2 3CLpro main protease in promoting viral infection and dissemination, we employed Inactive Catalytic Domain Capture and proteomics to identify host cell interactors and substrates in the interactome of 3CLpro. Of 259 3CLpro interactors, 145 were associated with the cytoskeleton and its organisation. We determined enzyme kinetic specificity constants for 139 3CLpro cut-sites in 43 interactors using a multiplex assay, identifying 29 efficiently-cleaved substrates and validating 13 as substrates in vitro, in SARS-CoV-2-infected human lung cells, and in COVID-19 post-mortem lungs. 3CLpro cleavage of adherens junction proteins initiated cytoskeletal rearrangement, activated zonular signalling, removed subcellular localization motifs to trigger translocation of nuclear TRIM28 and NUMA1 to adherens junctions, and YAP1 in the hippo pathway, to the nucleus. These cytoskeletal remodelling events rapidly generated tunnelling nanotubes connecting lung epithelial cells and containing virus colocalized with 3CLpro substrates for stealth trafficking of SARS-CoV-2 to distant cells.