Hi MFWills,
I'm assuming that you've already tried restoring your TM Backup using this support page:
https://support.apple.com/kb/PH18848?locale=en_GB
If it won't let you, then you'll need to make a Yosemite USB installer of at least 8GB. (a USB3 one will be much quicker).
First download the OS X 10.10 Yosemite installer from the Mac App Store in the Purchases section (you may need to hold alt/option while selecting Purchases. Once it's downloaded, just quit the installer and set the downloaded file's name to InstallOSXYosemite.app (make sure only .app, not .app.app)
The OS X Yosemite bootable USB flash drive will be ready for use and you can start the clean installation process.
How to Clean Install OS X Yosemite on Your Mac
- Ensure the bootable USB flash drive is plugged into your Mac and restart the computer. As soon as the start-up chime plays, press the Option key (Alt).
- Choose the USB drive on the start-up drive selection screen and hit Enter/Return on the keyboard.
- Wait until the Yosemite installer appears on screen. This could take a few seconds or minutes.
- Click on Disk Utility and then hit Continue.
- Click on Macintosh HD from the left hand-side pane and then hit Erase tab on the top-right portion of the window. Leave all the settings at their default values and then hit Erase button at the bottom right. This will wipe all files stored on your startup hard drive.
- When the erasing process is complete, return to the first screen where you selected Disk Utility. Now click on Install OS X and then hit Continue.
- Choose the Macintosh HD partition you erased in step 5 and then click Install.
You should now be on 10.10.4 or 10.10.5, either of which should allow you to restore to your Yosemite (10.10.4 or before) backup.
-Max