Qualifications improve your chances; they do not gate the job. Level 3 is the most recognised.
Level 3 most recognisedOpen entry
Your route in
You can start without qualifications - direct entry/volunteering; a qualification improves your chances
1
Qualifications employers recognise (Level 3)
Any of these strengthens your application.
2
Where this can take you
With experience, you could become a senior or principal conservator, managing a collections team. You may have more opportunities for career development if you join a larger institution, like a national gallery or museum. Other options include: research and teaching in higher education, working with a commercial conservation company and setting up as a freelance specialist, doing contract work on projects.