As stated in the widget"s documentation:
"If you add a layer that has a spatial reference different from that of your basemap, especially the tile layers, the added layer might not be visible." WAB Add Data widget
I was hoping that this limitation would eventually be overcome? Unfortunately, we cannot convince all councils (or even different countries) to publish their data online using the Web Mercator - they use local projections - and when you are working for the whole country or a very large area, it means that the data sourced from a council/country using another projection cannot be added using the Add Data widget. Which means that all the layers need to be added as default layers within the web map. Such a map becomes slow and unresponsive even if visibility, extent, labels, and symbology are configured for better performance.
The Add Data widget is a great and powerful tool, and I hope this limitation will be temporary.