nginx переписывают, изменяют параметр

did you read if is evil?

it still returns only JSON without the wrapper.

what is the wrapper here for you? the first/the second if or both?

EDIT:

i'd do simple debugging to test, if your regex is valid (increase log-level to see whats happening)(solution is untested)

server {
        listen          80;
        server_name     api.example.com;
        location /api/ {


                rewrite ^/(.*)callback=[^&]*(.*)$ /testurl/$1/$2 last;  
        }

        location /testurl  {
           return 200;
        }


}

after this step i'd put in the 2nd IF - statement, and if that works, the first one finally

how does your url looks like with that callback-parameters?