How can I access an activated child route's data from the parent route's component?

First child will give you access to data

constructor(route: ActivatedRoute) {
  route.url.subscribe(() => {
    console.log(route.snapshot.firstChild.data);
   });
}

Working with Angular 6, I managed to get the current route data from a parent component, with the following code:

I've configured the Router with the Extra options to inherit the parent routes data:

@NgModule({
  imports: [
    RouterModule.forRoot(routes, {
      initialNavigation: 'enabled',
      paramsInheritanceStrategy: 'always'
    }),
  ...
})
export class AppModule {}

Then in my parent component, I was able to see the data changes with:

import { ActivatedRoute, ActivatedRouteSnapshot, NavigationEnd, Router } from '@angular/router';
import { Subscription } from 'rxjs';
import { filter, map } from 'rxjs/operators';

subs: Array<Subscription> = [];

constructor(private router: Router, private route: ActivatedRoute) {
  this.subs[0] = this.router.events
    .pipe(
      filter(event => event instanceof NavigationEnd),
      map(() => this.route.snapshot),
      map(route => {
        while (route.firstChild) {
          route = route.firstChild;
        }
        return route;
      })
    )
    .subscribe((route: ActivatedRouteSnapshot) => {
      console.log(route.data);
    });
}

ngOnDestroy() {
  this.subs.forEach(s => s.unsubscribe());
}

Enjoy!

In Angular 8 work with:

data: any;

constructor(route: ActivatedRoute) {
  this.data = route.snapshot.firstChild.data;
}