var
 ButtonHandle:THandle;

function EnumChildProc(wnd:hwnd; lParam:integer):boolean; stdcall;
var
 buf:array[0..255] of char;
begin
 GetWindowText(wnd, buf, 255);
 if Lstrcmpi('Continue >>>', buf) = 0 then
  begin
   ButtonHandle := wnd;
   result := false;
  end
 else
  begin
   ButtonHandle := 0;
   result := true;
  end;
end;


procedure TForm1.Button2Click(Sender: TObject);
var
 WindowHandle:THandle;
begin
 WindowHandle := FindWindow(nil,'Form1'); // тут свои параметры укажиш
 if WindowHandle <> 0 then
 begin
  EnumChildWindows(WindowHandle, @EnumChildProc, 0);
  ShowMessage('Button Handle = '+inttohex(ButtonHandle, 8));
 end;
end;