From an undergrad school of Math to a grad school of Computing just to learn Computable Analysis — What are the consequences?
Computable Analysis as a subject sits on the border of mathematics and computer science, and correspondingly, you can find researchers from this area in either kind of department. There is not much correlation between whether a given researcher works on the more CS or on the more math aspects of computable analysis and what department they are found in. Either kind of grad school will have courses which are helpful in becoming an expert in computable analysis, and others that aren't. As such I would not put much weight on whether a department is CS or math in figuring out whether it is the right place for you to pursue computable analysis as a subject.

As computable analysis is a rather niche subject, you'll only find it in select places. In Korea, KAIST is indeed the place to go for it. If you do want to do computable analysis at a math department, you could eg go to Japan instead. A source for who is active in this area, you can consult 
http://cca-net.de/members.php
although this list is not that terribly up-to-date.