I'm really not sure how I missed it, but the vSphere 6 update finally brings the Update Manager components into the web interface!
The upgrade is pretty easy to, if you have 6.0 U1 already installed - as you can now use the VAMI interface (on port 5480) to run the upgrade. Simply go to https://vcenter:5480, login as your root user and go to Update. Then click the Check for Updates, and Install Updates options.... and sit back and wait.
You also need to upgrade the Update Manager component; now this is actually not as simple as I'd hoped - you need to download the full windows installation package and grab just the Update Manager installer from it. At least it completes an upgrade pretty happily!