Angular Lifecycle hooks are very powerful, in order to achieve what you want, you can simply use the OnInit lifecycle hook. Angular is reading your template file and running your component in a vertical way (from top to down). So, you just have to do whatever you want in your last component's ngOnInit method.
If your components are used in different parts of your application, you can check that the parent component is the one you want by using the @Host decorator (see the answer of @trichetriche).
I created a stack blitz where there's 3 components used in the app component (parent) in this order:
<hello name="{{ name }}"></hello>
<app-child></app-child>
So the AppComponent OnInit hook will fire first, and after that the HelloComponent OnInit and finally the ChildComponent.
See it in action on the console: https://stackblitz.com/edit/angular-62xjux
EDIT: You can use the ngAfterViewChecked
as the documentation specifies:
Respond after Angular checks the component's views and child views /
the view that a directive is in.
Called after the ngAfterViewInit and every subsequent
ngAfterContentChecked().