The following very short snippet shows you how to set/clear the network activity indicator on iOS.

uses
  iOSapi.UIKit;

function SharedApplication: UIApplication;
begin
  Result := TUIApplication.Wrap(TUIApplication.OCClass.sharedApplication);
end;

procedure TForm2.Button1Click(Sender: TObject);
begin
  SharedApplication.setNetworkActivityIndicatorVisible(not SharedApplication.isNetworkActivityIndicatorVisible);
end;