MacOS – Cannot update from OS X 10.11 to 10.11.1 – Stuck in loop

You mean you are still in App Store; you haven't actually launched the Install OS X El Capitan file yet [or it hasn't auto-launched]?

If the installer isn't yet running, you're safe to cancel it.

My App Store download errored at the last second & when I clicked the Update button again it suddenly realised everything was in fact fine.
Check if you have the installer app in your Applications folder. If you do, then it actually downloaded OK & you ought to be able to double-click it to continue.

As always on a major update, make sure your backup is up to date; & as a double safety-net, remove or disconnect your Time Machine drive until you are happy all went well with the update.

This is the kind of 'random' error I would entrust to Onyx [freeware] to tackle.

Run the Utility, supply your admin password, then allow it to run the disk check.

After it has finished, go to Maintenance & the 2nd tab, Permissions. Run the perms check.

If that is not sufficient to fix it, then try the default options in the last tab, Rebuilding.
Reboot.

Navigate to ~/Library/Saved Application State/com.apple.appstore.savedState & delete the contents of that folder, not the folder itself.

^{add a comment below if it still doesn't work}