I encountered this problem updating from 1.5.6 to 1.5.11. I think I was too fast doing things when it started, because I was closing Cursor forgetting it was still updating. Then I opened it again and see it installing. It got messy and I will keep story short, but yea, I had that same error about other process locking files.
In the end, I closed Cursor and tried to kill some background process called Cursor, but task manager said Access denied. Then I rebooted computer, launched Cursor, took Check update from Help menu, and ignored when it tried to offer update at that small hint bubble at left bottom corner. At some point the Help menu item changed to restart, and when I did choose that, then the update went through.