How should I bring my pet from the UK to Korea?

Question

Hello, I am a foreigner currently considering moving from the UK to Korea. Recently, my mother has been trying to suddenly rehome or dispose of my childhood friend, a cat, which is causing me a lot of worry.

I am looking into temporarily leaving the cat at a kennel or pet hotel in London, but I am curious about what procedures are needed if I eventually want to bring it to Korea. How should I go about vaccinations and preparing documents? Would it be easier to bring it over myself to Korea?

If you have any advice on good places in London where I can temporarily leave my cat, or information on the process of bringing a pet to Korea, please share. I am in a very stressful situation and really need your help. Thank you!