With new Angular Http, one can try the following in the Service code.

  downloadLink(): Observable<HttpResponse<Blob>> {
    return this.http.get<Blob>(this.someURL, {
      observe: 'response',
      responseType: 'blob' as 'json'
    });
  }

And use the above as

 this.someService
  .downloadLink()
  .subscribe(
    (resp: HttpResponse<Blob>) => {
      console.log(resp.headers.get('content-disposition'));
      data = resp.body
    });

Also, on the server side, one needs to set the following header in response. 'Access-Control-Expose-Headers': 'Content-Disposition'

Like in Java Spring Boot one can do the same using

    final HttpHeaders headers = new HttpHeaders();
    headers.add(HttpHeaders.CONTENT_DISPOSITION, "attachment; filename=1.xlsx");
    headers.add(HttpHeaders.ACCESS_CONTROL_EXPOSE_HEADERS, HttpHeaders.CONTENT_DISPOSITION);

To those complaining that the best solution is not working, and only content-type is in the headers, you need to set 'Access-Control-Expose-Headers': 'Content-Disposition' ON SERVER SIDE. I am using asp.net core, then I have to do the following.

app.UseCors(builder =>
                builder.WithOrigins(originsAllowed.ToArray())
                    .AllowAnyMethod()
                    .AllowAnyHeader()
                    .WithExposedHeaders("Content-Disposition")

You can try this kind of code here.

Note: Do not use map

this.http.post(URL, DATA)
  .subscribe((res) => {
     var headers = res.headers;
     console.log(headers); //<--- Check log for content disposition
     var contentDisposition = headers.get('content-disposition');
});