I'm trying to update to Lightroom CC and it says it has downloaded but the new version is not available if I log out or shut down - it doesn't update from Lightroom 5

If you subscribed to creative cloud, you first need to install the Adobe Creative Cloud app:Creative Cloud Help | Creative Cloud for desktop.

Be sure to sign in to Adobe using the same adobe id (email) you subscribed with.

If you can't install an app, please check that your system meets the minimum requirements. Apps will not show up if the system can not support them.  System requirements | Creative Cloud 

  Since Lightroom 2015 CC is a new product, it will not show as an update in the Adobe Creative Cloud desktop app. Instead, press the  install button next to Lightroom in the Creative Cloud  Application manager. If you don't see "install" by Lightroom in the application manager, please try signing out and back in: sign in and sign out of the Creative Cloud desktop app

If you are trying to upgrade from Lightroom 5 to Lightroom 6 (not creative cloud), then see this link:  Products  .  Scroll down to lightroom and click "buy". Click on the selectable text by  " I want to: "and select "upgrade".

Pattie