WHOA!!
I did NOT say processing takes 3 months. I'm saying wait at least 3 months for an AOR (Acknowledgement of receipt) of your application, before worrying.
NS does not process applications. If you are in Canada, your application would be processed in Ottawa, so yes, as you've seen, around 15 months.
Yes you can do an upfront medical. HOWEVER, the medical is only valid for 12 months. If you do it now, you risk it expiring & needing to do another (= more $$$).
In conclusion, it's highly unlikely you can speed up the process for whatever reason, personal or otherwise.