I'm Korean Librarian. My undergraduate major is Library and Information Sceince. And I've MLIS by Korea univ. I'm working Special Library now. But I want to…

I would advice you to look at CILIP's (former British Library Association) website. https://www.cilip.org.uk/

There is a whole section on how to start a library career in the UK under the Careers-section. There is also information on qualifications needed in the UK. Good luck with your job search!