Stu did not develop the iOS app. It is from the Grand Poobah. Grand has Stu's URL in the app to test the Internet connection. The theory is if they will connect to Stu's site, they should connect to your site, if you have the settings properly sorted on your router.
From The Grand Poobah himself, earlier in this thread:
There are a few things that you can check although I suspect you've already done this:
1. Change the settings on your iOS device to "Local" to verify that you can get any data.
2. Change the settings on your iOS device to the trixology-test address to verify that you can get data over the Internet.
3. Inspect the router administration page to verify that a private fixed IP address is still assigned to your host computer.
4. Inspect the host computer Network settings panel to verify that the private fixed IP address is being used.
5. Inspect the router settings to verify that the port range 49250 to 49254 is assigned the the private fixed IP address.
Number 3 is especially important if you have moved your WeatherCat installation to a new computer.
If you have already done all of these and still no joy, which is what happened to me, you can try reinstalling Yosemite. Something does not play nice for some issuers only between OS X and the iOS app. Nobody knows what it is. For me, rebooting the machine on which WeatherCat runs will periodically cause the iOS app to stop connecting. Several more reboots will eventually cause the app to start working again. For me, during the times when the app will not connect via the Internet, it always connects locally.
Since this started happening to me, I have noticed that it's all or nothing. In other words, if the iPhone app won't connect, neither will the iPad app. If one of them will connect, they will both connect.
During the times when I can't get the iOS app to connect, I use the WU app to see my weather data, or simply go to my site on my iOS device. But what I love about the iOS app, when it's working, is the ability to see the WeatherCat Core data: Station Comms, Good Data and Comms Errors.