I'm looking for an Open Source Android app for routing through our own network (mandatory). If possible, able to work offline.
Check out this link.
Heaps of options, and it does mention open source options. All will use OpenStreetMaps as the map and routing.
The Maps.Me solution looks pretty good - https://wiki.openstreetmap.org/wiki/MAPS.ME Navit seems to be more focused on in-car navigation, which might be more suitable depending upon what you are looking for.