Fix Mount routes being silently dropped in include_router

When a sub-application is mounted on an APIRouter via router.mount() and
the router is later included with app.include_router(), the Mount routes
were silently dropped because include_router() had no handler for
routing.Mount objects. Added an elif branch that creates a new Mount with
the combined prefix and appends it to self.routes.

Closes #10180

fastapi/routing.py

@@ -1548,6 +1548,14 @@ class APIRouter(routing.Router):
                 self.add_websocket_route(
                     prefix + route.path, route.endpoint, name=route.name
                 )
+            elif isinstance(route, Mount):
+                self.routes.append(
+                    Mount(
+                        prefix + route.path,
+                        app=route.app,
+                        name=route.name,
+                    )
+                )
         for handler in router.on_startup:
             self.add_event_handler("startup", handler)
         for handler in router.on_shutdown:

tests/test_mount_sub_app_router.py

@@ -0,0 +1,58 @@
+from fastapi import APIRouter, FastAPI
+from fastapi.testclient import TestClient
+from starlette.routing import Mount
+
+
+def test_sub_app_mounted_on_router_is_accessible():
+    """Sub-apps mounted on an APIRouter should be reachable after include_router."""
+    sub_app = FastAPI()
+
+    @sub_app.get("/status")
+    def sub_status():
+        return {"status": "ok"}
+
+    router = APIRouter()
+    router.mount("/sub", sub_app)
+
+    app = FastAPI()
+    app.include_router(router)
+
+    client = TestClient(app)
+    response = client.get("/sub/status")
+    assert response.status_code == 200
+    assert response.json() == {"status": "ok"}
+
+
+def test_sub_app_with_router_prefix():
+    """The include_router prefix should be prepended to the mount path."""
+    sub_app = FastAPI()
+
+    @sub_app.get("/info")
+    def sub_info():
+        return {"info": "value"}
+
+    router = APIRouter()
+    router.mount("/sub", sub_app)
+
+    app = FastAPI()
+    app.include_router(router, prefix="/api")
+
+    client = TestClient(app)
+    response = client.get("/api/sub/info")
+    assert response.status_code == 200
+    assert response.json() == {"info": "value"}
+
+
+def test_mount_preserved_in_routes():
+    """Verify the Mount object ends up in app.routes."""
+    sub_app = FastAPI()
+    router = APIRouter()
+    router.mount("/sub", sub_app)
+
+    app = FastAPI()
+    app.include_router(router)
+
+    mount_routes = [
+        r for r in app.routes if isinstance(r, Mount) and "/sub" in r.path
+    ]
+    assert len(mount_routes) == 1