Is British attention really exclusive? Especially because, assuming America settles down, it runs itself.
Won't be totally exclusive, no. Especially since India was a very important trading area. However Britain maintaining control of the 13 states and western territories will be a big factor making changes. Especially since that will mean when the wheels come off the Spanish empire in Latin America, as it almost certainly will do, its bordering British North America.
Also a lot would depend on the circumstances. You would need something like the basically ASB scenario in "For Want of a Nail" where a British North America virtually ignores events elsewhere in the world, even when Britain on the verge of invasion for it to have no real impact. Let alone the question, for better or worse, of the interactions of the British and American economies and societies. Also, having nearly lost the colonies once Britain is likely, for a while at least, to be paying close attention to events there, whether intrusive or appeasing in approach.
Basically; I'm envisioning something like a permanent consulate under Bernadotte.
Why a greater military threat? Because a France that encompasses the Rhineland and Low Countries, under competent leadership, simply is a greater threat.
If it stays stable and encompassed those lands it would be a greater economic and potential military threat. However would it be able to? Such gains not only threatens Britain but with the spread of nationalism will cause problems in the German speaking lands as well as probably the low countries.
Also if there's no open war, or its greatly curtailed, you avoid the huge expenses of the Napoleonic conflicts. A lot less death and destruction, disruption of trade and massive subsidies to continental allies.
Furthermore is the French mega-state being maintained on a high military footing or not? If not its markedly less of a threat. If it is then its going to struggle to fund things without looting of the neighbours as in Napoleon's time.
Not saying its impossible but needs some balancing of the various forces at work. A reasonably stable and successful moderate consulate could well prompt unrest in Britain simply by posing an attractive alternative. There were strong forces in Britain favourable to the early republic until it went off the rails and started proving too unstable and expansionist.
Would India change? I'm actually surprised; I would have thought you'd think after Plassey India's fate was sealed, for the most part.
Plassey set the EIC on the road to territorial dominance but it was still fundlementally a trading company looking to maintain stability and minimise costs. By 1763 French direct military power in India was largely destroyed but there were still a lot of native states which if they have found some stable situation may well have lasted quite a while. [Or possibly one having emerged to form a new empire over much of the area. The latter might have led to a clash with British power as it would probably have sought to interfer in its trade or territory. However this needed necessarily have been the case.
Probably still likely that India would have come under increasing British political/military dominance but in what way and during what time period could change drastically.
A small note on the mutiny. From what I've read it was initially largely a mutiny, by some elements of the sepoy forces. Probably largely due to the fact that their economic and social position had deterrated as the dominance of the EIC reduced instability and the need for large numbers of troops. [Interesting that one of the 1st things the mutineers generally imposed on the various native rulers who often somewhat reluctantly 'rebelled' was a sizeable increase in their salaries. Unrest was also made more likely by the growing separation between the sepoys and their officers and as Rob C says by the growing interference by the British government. Some nasty attrocties occurred, most infamously the two Carnpore massacres, which by most accounts the British/Loyalist forces often retaliated against, and which worsened relations on both sides. However by no means certain that even with a politically dominant EIC it would have occurred other than possibly a few isolated clashes and uprisings.
Steve