So I managed to get this to work! So i'll explain the steps here for anyone else finding themselves in the same situation.
If you have Avast Password Manager on a machine and you were *not* syncing with an Avast account, you *can* transfer the passwords from one hard drive to another machine and access the passwords!
Firstly, you need to get the passwords of the machine thats broken, so you need to access the machines hard drive.
Then, copy the passwords from the following folder location...
C:\Users\<your username>\AppData\Local\Avast Software\APM
Inside there you should see a folder and a file (maybe more than one) with your username in the name, copy all of those somewhere.
Then on the new PC, you have to do the following steps with Avast *uninstalled*. You won't be able to copy those files into the new harddrive at the same location while Avast is installed, the AppData\Local\Avast Software folder will be blocked.
So, uninstall Avast.
Then copy over those files into the same location...
C:\Users\<your username>\AppData\Local\Avast Software\APM
You *may* have to renamed them too. If your username on the new machine is different to the previous one, rename both the folder and files that contained your username with the new one.
Reinstall Avast.
Open up password manager and it should prompt you for the Master Password, input that and you're good!