I can't say much about the CS side, but I can write about what it would mean to study computation on the mathematics side. Perhaps if someone else writes about the CS side, the OP can compare.
In mathematics, computability is studied as part of mathematical logic. The key initial concept is Turing computability, but the focus is just as much on non-computable objects. So the structure of the Turing degrees is a key topic at first. Within computability theory, there are many areas: "classical" computability, higher computability, Reverse Mathematics, computable analysis, and crossover with areas such as proof theory, effective model theory, and effective descriptive set theory. Topics such as computational complexity theory, compiler/language theory, and automata are not studied as often in mathematics departments.
The methods used are very mathematical. Mathematical computability theorists tend to focus much more on proving results than on implementing anything. For your qualifying exams, you will need to learn several other basic areas of mathematics at an introductory graduate level, such as real analysis and/or abstract algebra. An undergraduate degree in mathematics, or very good mathematical preparation otherwise, is required to be admitted to a PhD program in math.
The field of computability is not extremely large, which can make both finding a PhD program and finding an academic job more challenging. Essentially, mathematical logic as a whole is only as big as a subfield of many other areas of logic, and then computability theory is only a part of mathematical logic.