Is it possible to instantly deliver a web push notification to notify a phone call or to deliver any other time-sensitive notification?
Web push notifications are delivered in near real time. This is great for news or messaging applications: most notifications are delivered instantly within seconds and even if some notifications are displayed by the phone a few minutes later it's not a problem.
In particular many mobile operating systems, like Android, use different strategies to save the battery power and this can slow down the delivery of notifications. For example they can delay the processing of non urgent messages by asking to the push service to deliver only the urgent messages.
When you send a push message from your application server to the push service, using the Web Push protocol, you can specify an Urgency header. This header allows you to specify if a message should be delivered in real time (e.g. incoming phone call) or can be delivered some time later (e.g. news).
If you send push notifications from your web app using Pushpad, you can set the priority of notifications.
Note however that at the time of writing some browsers cannot wake up the OS from deep sleep and this may create problems (i.e. the delivery is not in real time) in some circumstances. In particular you can read this discussion in the Chromium bug tracker: Issue 777106: SW: push messages not waking Android 8.0 from deep sleep.