How could I become a mathematician with only a bachelor's degree in computer science?
Undergraduate math has a wide variety of free or cheap resources online.  Youtube, udemy, and khan academy are your best friend. You will quickly learn how to spot a good teacher / course.  Some books are public and many are available at libraries or online. And of course stackexchange is an option when you feel stuck. As you said, you have a number of years before you would try to make this transition.

This is coming from someone who went from being a teacher to learning python and getting a job at more than double the pay with just 3 months of online learning (mostly udemy and youtube) and a 3 month internship after that

As you learn more and gain a stronger grasp on undergraduate mathematics, you will begin to answer your own question about what topic you really want to pursue.  

Although, even now, it does not hurt to send out a few emails and ask professors about topics.  The worst that can happen is them ignoring you.  You might "waste" 3 minutes of 100 professors time but a single answer could inspire you or help you make a more informed choice about your future careers.  I think it is worth is.

If you are unsure the topics you should study on and the recommended order, find transcripts or graduation requirements for a mathematics degree in a school you respect. It might take a little digging, but finding exactly what you would need to take at x school for y degree is not impossible (heck, send emails to counselors there and just ask if you cant find it easily online)

for an undergraduate reference 

https://webapp4.asu.edu/programs/t5/roadmaps/ASU00/LAMATBS/null/ALL/2018?init=false&amp;nopassive=true

Even vague terms like "upper division depth course" can be clicked on there and it leads to a list with descriptions of what they are and what is taught.

I found this by just googling 

course requirements for math degree ASU

similar searches should yield fairly similar hopefully helpful results