Why would Britain escalate the conflict,what justification does Britain have to intervene in the defense of Indian territory and why would India break with its policy of neutrality and allow it's former colonizer on it's soil.
In this case it would be Indonesia that was escalating.
Britain was a Guarantor for India and for that matter Pakistan and East Pakistan.
In the case of a kick up between them both - the UK and Commonwealth can only really wring their hands and try to calm the situation (which was actively being done by both nations already in trying to limit the conflict to that one region)
In the case of Indonesia invading those Islands....Well this is where Sabres can be rattled and the mournful primordial howl of Vulcans would be heard above Jakarta to refocus attitudes.
As for timelines - Note that right up to 1st Sept - the Indian Navy's primary concern had been the defence of those Islands (which had become their sole responsibility) against any possible adventurism by Indonesian forces and much of the operational fleet not undergoing planned refit was based their - it was only after the shelling of Dwarka that they redeployed to the West coast.
It was only after the Indian counter attack (6th Sept) that Pakistan sought out help from Indonisia (and others) and Air Marshal (Rtr) Asghar Khan of the PAF visited the President Soekarno and Admiral Martadinata of Indonesia
The Admiral proposed increasing patrols of the area and conduct aerial reconnaissance missions to determine what Indian Ships were in the area which implies they had no clue as to Indian Navy dispositions
The other major issue regarding conducting operations in the area at this time is that this is the latter half of the Monsoon season - one of the reasons why INS Mysore and the Type 41 Frigates operated there was that smaller vessels were ineffective at this time due to the weather so given that the Indonesian navy comprised of 1 Cruiser and half a dozen odd DDs at various states of readiness with no real amphibious capabilities
So there is maybe a tiny window of opportunity - that is only known today after completing the jigsaw - and would not have been known by any one faction at the time - for Indonesian to invade the Islands - which is about 2 or 3 weeks in Sept