The only data it would use is the downloading of the page on the browser that may indicate that it's being updated or something (I never used the WordPress updater so don't expect me to know a lot on this part of the subject).
As stated by many in this thread (and I'm sure that you've picked it up by now), PHP is being used to update WordPress. As PHP is a server-side language, it is bound to be a server-side operation. Yes, PHP is able to download files without any involvement of uploads.
If it was to be an upload (from the client side), it would request you to save a file, ask for you to pick a file (assuming that there's no JavaScript pre-specifing the path) and then on PC browsers (should) tell you that it's uploading a file along with the percentage, like on Chrome it says 'Uploading (xx%)' in the bottom left corner.
I hope my post makes things more clear.