If I've decided to go to Grad. School for Mathematics... a little late?
The next Mathematics Subject GRE is in April. If I want to get into a mathematics graduate program, have I missed my shot?


Yes, it will hurt your chances for this year, but you might still be admitted.  Admissions committees can sometimes be flexible about requirements like this if you give a good explanation, so it could still be worth applying (but what the chances are depend on the department and perhaps even the specific committee members).


  How viable of an option would it be to switch from a physics grad. program into a mathematics one in the same university?


As a general rule, being admitted to one department does not make it easy to transfer to another.  If you have good interactions with the mathematics faculty, then that might help some, but you would still have to apply in the usual way (and the chances of admission would in general be comparable to what they might have been if you had not been in the physics department already).

[Added in edit:] On the other hand, you may be able to accomplish your goals in either department.  You can certainly talk/work with faculty from both departments, at least informally and perhaps through some sort of formal arrangement.  If you do work in the intersection of both fields, in the long run it won't matter very much which department your degree was from.  Mainly, it's a matter of which requirements you'll have to complete in the short term (will you take more physics classes or more math classes?).