The alerts are fairly quick but not instant. Sometimes around 10 secs, sometimes longer. Yes, they are push notifications. They are coming through OK, just not in a timely manner.
OK, I think I just figured this out. While outside testing the triggers and alerts my WiFi connection was dropping and reconnecting. It's probably the connection time that delays the alerts. After further testing last night the alerts are coming through much quicker (just a couple of seconds).