The scenario has two main branches: Milne either keeps the Mediterranean force together or detaches Beatty with the ships historically assigned to Troubridge. British battlecruisers had already shadowed Goeben and Breslau before war was declared. They could observe the German squadron but had no authority to open fire, and even Beatty was unlikely to act differently under that constraint. Aggressive pursuit would still have to respect Britain's position before the declaration of war.
Notable Point · 35:58-37:17
Why pre-war shadowing did not become an attack
British battlecruisers could shadow Goeben and Breslau before the declaration of war, but even Beatty's aggression would not by itself authorise an attack.