I transitioned from Mathematics for Computer Science Ph.D. as some of my interests changed course.
Bulk up on Algorithms and Data Structures. Generally, there is some Proficiency Examination that attempts to evaluate your fundamental understanding of the subject matter before you are accepted to a department at the doctorate student level.
If you were in a Ph.D. Mathematics program then you most likely had to take this in Algebra, Analysis, Probability, etc. When I switched, I had to take one in the Computer Science department whereby the material covered was, C++ , Data Structures, Discrete Mathematics, and Operating Systems. Good luck, it looks as though you can make the transition seamlessly ...