The buck stops with the Captain. Every time.
He would most likely have consulted Operations Control and presented them with what had happened and the options that were available to him (ie. I can continue to LHR, I can continue to the East Coast, I can hang about for a couple of hours and dump fuel). What would you like me to do?
If he had said to Ops Control, "I am continuing", or "I am diverting", they would have had to accept it. It's all in accordance with SOPs.
Had it been a "catastrophic" engine failure such as "Fire" or "Severe Damage/Separation", something which degraded safety, I would venture that BA Ops Control would not have found out about the situation until the aeroplane was back on the ground at LAX.