Hi Maggie, There is no legally 'qualified' in the UK, but Level 4 is considered the minimum standard for a qualified counsellor within the profession. If someone has completed the Level 4 Diploma with all their required supervised hours then they're no longer a student (unless they're taking a further, additional course). The BACP only accepts people as registered members if they are qualified to at least Level 4 standard.
If they aren't on the BACP register when they claim to be then that's a red flag and something to raise with them. Do be aware though that someone can be BACP registered but not in the BACP directory which is an additional charged-for service. The register is available at https://www.bacp.co.uk/about-us/protecting-the-public/bacp-register/ so do double check. If they don't appear then you could ask them for their registration number in case they're registered under a different name, eg their married/maiden name. If you don't feel comfortable with them then there's nothing wrong with switching to a different counsellor even if they do appear.
The BACP directory (paid for adverts) is at https://www.bacp.co.uk/about-therapy/using-our-therapist-directory/