Resources for mathematicians hoping to learn more computer science
You're essentially asking for resources that will let you turn your existing knowledge of logic, recursion theory, and category theory into knowledge about theoretical computer science.I would suggest looking at realizability theory, especially via its connections to topos theory and categorical proof theory. 

Here are a handful of suggestions; my advice is to pick one and go into depth. With the exception of Taylor's book (which explains this), my suggestions assume you have been exposed to enough lambda calculus and category theory to have seen categorical interpretations of the simply-typed lambda calculus.


Paul Taylor's book Practical Foundations of Mathematics

IMO, this is probably the single best technical introduction to the relationship between logic, category theory and computation. It assumes almost no pre-requisites, but it gets into quite deep waters very quickly, and is sure to tax (and greatly improve) your mathematical maturity.  
Wesley Phoa's notes An Introduction to Fibrations, Topos Theory, the Effective Topos, and Modest Sets

These are some lecture notes that Wesley Phoa wrote up. If you are categorically fluent, then these notes offer a really fast path into understanding some of the most  important constructions in realizability and topos theory (namely, the construction of the effective topos).  
Bart Jacobs' book Categorical Logic and Type Theory

This is one of the definitive references on the categorical semantics of type theory. It is also very large. 


At the same time you are reading one of these books, I would advise downloading and learn how to use the Agda programming language. This language implements the sophisticated type theories described above, and IMO it is incredibly helpful to see how the often quite subtle semantic constructions cash out in type theory. 

Andrej Bauer can probably give you even better advice. Perhaps he can be persuaded to post. :)