Does designating a Data Protection Representative release the non-EU or non-UK companies from liability and responsability?

NO. The GDPR and UK GDPR clearly state that the designation of a Data Protection Representative does not affect the responsibility and liability of the non-EU and non-UK companies that fall within the scope of the GDPR and UK GDPR. The designation is without prejudice to legal actions which could be initiated against the non-EU and non-UK companies.