Are there Chinese approximations for the colloquial/slang English terms "snot rocket" or "bushman's blow"?
From the comments it seems like we've already established that blowing your nose is accurately translated as "擤鼻涕". As far as I can tell (from experience and further online research), there is no popular slang for doing so. The closest way to easily describe such action would be something like "不用纸擤鼻涕". 

However, I should point out that this action is in fact pretty stigmatized in the Chinese culture, especially in more developed regions. Chinese people are simply less self-conscious about what others might think of their actions, which is why you'll still see people doing it.