All luxury car companies
Given the number of talented automotive industry professionals in India, if they really wanted, I don't think it'd be too much of a stretch. After all, Mercedes has been building cars in the country for years, never mind that Tata owns JLR.
The question is why would anyone build an Indian luxury car? First off, if it were a new brand, it's important to note that the barriers to entry in the industry are extremely high. The only new brands that have succeeded are ones that were either created or bought out by larger, existing brands (Tesla being a notable exception).
Then there's the heritage problem. Indian cars, up until fairly recently were basically rebadged versions of foreign cars. Furthermore, homegrown Indian designed & built cars aren't up to global standards yet. Let's take a look at the Tata Indica for example. Even when it was brand-new, it was in the lower half of its segment (build, dynamics, performance) compared with the global competition. For a car-maker such as Tata, it would be a huge jump to all of a sudden start offering a world-class luxury car. Before the inception of Lexus, Toyota's models were at least class competitive. Same can be said of Hyundai introducing the Genesis line-up (I know Hyundai's dabbled in luxury cars in the past, but these were mostly rebadged Mitsubishis).
This then brings us to the question, why aren't Indian cars competitive yet? Simple, they're primarily being designed and engineered for the Indian market. While there is increasing demand for luxury cars, the overwhelming share of the market is looking for basic conveyance. Safety? Forget about it. We're talking about families here that are stepping up from a motorcycle. Indian car companies know they're target audience, and are catering to them rather than to the larger global market. At the end of the day, Indian cars are all extremely utilitarian, and that's shaped by market needs.
So where does that leave Indians shopping for luxury cars? With foreign cars. These people have more sophisticated tastes, are generally better educated, and are well-travelled. They have greater exposure to foreign brands, and prefer foreign cars. Of course, this is a gross generalization. Let's not forget though that there's always been belief in the subcontinent, that "foreign" or "imported" goods are better (even if they are the same brands).
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What is the most luxurious car company?
Matter of opinion, if you were looking for the most luxurious car ever, this could just fit your bill. The Maybach 62S Landaulet.