How to transition from theoretical mathematics to theoretical computer science
The answer to your question in "no": Most of people who call themselves Theoretical Computer Scientists have PhD in (T)CS. 

And yes, you need to have a CS background because the word says it itself: Theoretical Computer Science which is the study of what a computer can(not) do and with what resources.

Math is a vital part and on different fields of TCS different fields areas of math play important role (number theory, linear algebra, combinatorics. probability theory are by far the most used ones). Generally speaking, the maths used are rarely "obscure" or too abstract/pure.

As on the question of if this can be achieved outside formal education the answer is "depends" on the motivation, circumstances etc. It is really hard to answer that.