Maybe. You could probably get pretty far with atomically moving a symlink so that the filesystem view always looks at either all the old or all the new files.
However, even that doesn't handle in-flight requests that have their view of the files swapped out from under them. Yes, that's a small time window for an error to happen, but it's definitely not instantaneous.
The safer solution would be to update the server config to point at the new directory and reload the webserver, but now you're way past just uploading the new files.