Since you didn't mentioned about the phone model, XDA forums contains a large number of users and developers (5,575,771 Members) that are focused on Android development, flashing, upgrading, etc. It's the only place on the Internet where you could get help regarding phone downgrade and flashing.
As you have said, the backup is not recognizable on the newly updated version. This means that you will actually need to downgrade to the previous version of the Android. And this can only be achieved using custom tools that have the capability of flashing another version of the Android OS. On another hand, 3rd party developers are actually independent developers (regular people with developer skills) that have created tools to easily restore Android to a previous version since official companies don't offer these tools to end-users.
