I suspect that you are seeing the behaviour as described in your question from the development server, which cannot serve more than one request at a time. So when you put your breakpoint in the POST method, the dev server is effectively paused. To see this behaviour as intended, upload to App Engine.
Further, instead of polling using a GET HTTP request, try using App Engines Channel API. Using this, you can push a response back to the client directly from your task queue handler to notify that the process is complete (or whatever).