How long would it take to become proficient in machine learning for someone with a non-statistical mathematical background?
I think you already know enough applied mathematics to begin with. You can pick-up rest of it as required. 

One option is :


Start with an online course that provides high level overview of machine learning and types of algorithms (E.g.: https://www.coursera.org/learn/machine-learning)
Start applying the knowledge in real world problems as soon as possible. 
Learn various types of neural networks (deeplearning.ai is one place to start)
Apply knowledge to real world problems (Such as Audio/Video classification , Natural Language)
Get an internship 


This will take 5 - 6 months.