Basically you need to implement a load test
Implement the scenario(s) of your application usage by real users (or other systems) using a load testing tool
Execute the test against the "old" system and collect relevant metrics, the key ones are:
average response time (as well as percentiles)
throughput (number or requests per second)
number of ...
Check out Apptim. It was designed specifically for client side performance of native apps and can work for hybrid as well.
This is a good article that explains how to run a test with Apptim and can give you an idea if it is the right fit: https://blog.apptim.com/mobile-performance-testing/
So, there are three main components of mobile app performance:
When it comes to the backend, the things related to performance that are important to know when an app is under load are the server response times, database queries times, and the server’s resource usage.
This is where load testing with a tool like BlazeMeter ...