(Like many of my technology postings, this is really here for my own notes on problems I run into, the solutions I try and what works when. They are updated as I learn.)
After uninstalling Itunes with Revo Uninstaller, I followed these steps from Gheeda Al-Zubaidi.
To solve this issue:
1. I right clicked the installation file
2. Went to properties
3. Clicked unblock
Then:
1. From the start button, searched for "uac"
2. Clicked "Change User Account Settings"
3. And dragged the bar to "Never Notify" (needs restarting)
4. After restarting, I ran the installation as administrator again, and it worked!