Could Open Source have prevented the Volkswagen scandal?

Could it have? Sure.

Is it probable that it would have been prevented? No.

Look at the number of open-source projects that have had vulnerabilities in them for years. Open Source does not translate to "we know exactly what's going on in this code", it just means the source is available. Open source is not some magic catch-all for shitty or purposely misleading code.

Probably not. The same situation as with Android on mobile phones, tablets and TVs: locked in bootloader, proprietary peripheral drivers, proprietary applications to run everything.

It would have be the source to everything without any exceptions, down to bootloader, and then preventing producers to void your warranty when you have changed your software.