This app was mentioned in 4 comments, with an average of 1.50 upvotes
Pro Weather Alert on Android. I have it set up to only override silent mode for Tornado Warnings. They only alert me to when something is happening in my part of the county too.
So, there is https://play.google.com/store/apps/details?id=com.globalcanofworms.android.proweatheralert , they have simple weather alert for free. I decided to pay for it after using it for one season. It is close to what your asking for. There is gps mode that takes your current location into account as it's checking for alerts, once watches or warning issued by nws, you get a notification. You can also configure dedicated counties or cities by zip. It is highly customizable, almost too much in fact. You can set urgency levels, quiet times, rules to over ride you phones silent mode, specify events, specify how the phone alerts you (light/vibrate/sound) just once or constant. I like it because it is light weight, extremely customizable, so far very reliable and cheap. My only complaint, which isn't really the fault with the app is that nws updates watches and warning so often that on some high tor% days it becomes more annoying than useful. However, being informed and annoyed is better than ignorant and dead. There maybe better options out there, but I've had no issues with this app. I've used it for about 3 years now, maybe longer.
Install Pro Weather Alert. It actually uses your location and gives you alerts that are tailored as such. For example, instead of getting a state- or county-wide alert, you will get an alert of events that will directly affect you.
The best part is the tornado alerts. It will alert you if you are directly in the path of a tornado. It will also show the path of the tornado on a map with probability markers so you can see if you're dead center ... or on the outskirts.
We were woken up in the middle of the night by this app due to a tornado coming right at us and we were able to get all the kids and take shelter with just minutes to spare. Had we not been woken up ... maybe not such a good outcome.
It's an amazing app and is totally underrated IMO.
You all need to install this app:
https://play.google.com/store/apps/details?id=com.globalcanofworms.android.proweatheralert&hl=en
If there is an actual tornado warning, and you're in the path, it will make your phone go nuts.
The best part, however, is you can click into the alert, and then it has a map option, and you can click into it and literally see the exact location and predicted path of a tornado.
Last year we were right in the path, and took cover. This year it was 20 miles away and leaving the area.
I can't recommend this app enough. Knowing where the potential for a tornado actually is located totally helps.