Is it possible to automate screen reader testing using Jaws and NVDA across all the browser platforms for checking web accessibility ?
Short answer is no, you cannot effectively do this.
However there are lots of automated tools that will identify around 40% of accessibility errors, such as empty links etc.
The reason you need to test manually is that they cannot test for things like logical tab order, keyboard traps etc.
Axe do a decent plugin for Google Chrome that is reasonably good as a starting point. Look in the Play store for that.
Other than that, learn how to use a screen reader (it doesn't take long) and test the site manually, you will pick up accessibility issues far quicker and learn more than trying to follow rules etc.