How to call function by order without calling setTimeout ? using Angular

allData() performs an async operation so you need to call the second function when the getContactPerson() returns a value. In your case you need to call it in the next callback. Try to move the second function call inside the callback, as following:

allData(){ 
   this.accountsService
    .getContactPerson()
    .subscribe( 
       (data: any) => { 
         console.log(data)
         /* DO SOME STUFF HERE */

         this.lastAddedItem(this.contacts, contact);

       }, 
       (err: Error) => console.log(err) 
     ) 
} 


this.allData()

UPDATE

If you want to use the same function in onInit and after another event such as a button click, you can refactor you code to return an Observable and use it in multiple places.

allData = () => this.accountsService.getContactPerson();

ngOnInit(){
   this.allData().subscribe( 
       (data: any) => { /* DO SOME STUFF HERE */ }, 
       (err: Error) => console.log(err) 
     ) 
}

refreshList = () => {
   this.allData().subscribe( 
       (data: any) => { 

         /* DO SOME OTHER STUFF HERE */

         this.lastAddedItem(this.contacts, contact);

       }, 
       (err: Error) => console.log(err) 
     ) 
}


// This is just an example for an async call
anotherAsyncCall = () => {
   this.allData()
       .pipe(
          switchMap((result) => this.contactService.makeAsyncCall())
       )
       .subscribe()
}

This is a classic case for using switchMap operator. First of all, do not subscribe in allData() method, instead just return the observable:

allData() { 
    return this.accountsService.getContactPerson();
} 

then, use switchMap where you are calling allData():

ngOnInit() {
    this.allData().pipe(
        switchMap(data => {
            // Do what you want to with the data. 
            // I am assuming that the data is either assigned to contacts 
            // or contact ... but I am not sure. 

            // Return if this method also returns an observable.
            return this.lastAddedItem(this.contacts, contact);

            // otherwise comment the above line & uncomment the below line:
            // this.lastAddedItem(this.contacts, contact);
        }
    ).subscribe( ... );

I believe that you are getting observable from this this.accountsService.getContactPerson() method, so I will suggest you please try this approach.

async ngOnInit() {
   await this.loadAllData();
   this.lastAddedItem(this.contacts, contact);
}

public async loadAllData() {
    try {
        let allDataResponse$ = this.accountsService.getContactPerson();
        let allData = await lastValueFrom(allDataResponse$);
    
    } catch(e) {
       console.log(e);
    }
}