Recruitment consultant roles are generally open to graduates regardless of degree subject. Employers typically value drive, experience, skills and abilities as well as the determination to succeed over specific qualifications. Applying to an agency that specialises in a market or industry related to your degree subject, for example IT or engineering, may increase your chances.
Entry with an HND or foundation degree is possible. Entry without a degree is possible for those with relevant experience, skills and abilities.
Postgraduate study is not essential. You may find that relevant experience and skills are more useful.