We'll ignore the various tussles of the British and French over colonies - the smaller countries got what the French and British weren't so bothered about.
From the point of view of the Imperialists, India was the Empire, it outweighed everywhere else in significance and was also key to the security of trade routes in both south Asia and central Asia (where Russia was seen to be expanding). A threat to the 'jewel in the Crown' of the Empire was not to be entertained under any circumstances and Russia's tendency to territorial and influence expansion was a fact of European and Asian history.
The British and French had some early tussles over colonies, although it seems by the mid-19th century, they had gotten over it and seemed willing to co-exist with each other. Britain and France were on the same side with each other in the Crimean War.
Russia's expansion into Central Asia was a gradual, incremental process which took place over the course of centuries, as the flagging Mongol Empire fell apart, piece by piece. The Russians had a presence in Asia while England and Scotland were still fighting each other and before Britain had even existed as a unified kingdom - long before they staked any claims on India. I'm sure the British were quite well aware of the boundaries of Russia when they showed up in India (although they weren't the only European power vying for a piece of India).
Then there's also the matter of the distance and terrain. A cursory look at a topographical map of Asia would show large mountain ranges dividing Central Asia and the territory associated with what was once known as British India. Trying to send an army from the European part of Russia across the deserts of Central Asia, and then across the mountain ranges in and around Afghanistan, just to get to India, would be a colossal undertaking and a logistical nightmare. The British army would be there to be meet them, along with many local volunteers. It would be a complete disaster for the Russians - far worse than the debacle they endured against the Japanese.
And you're saying the British were genuinely worried about this possibility? Seriously? Based on what?
Of course, if they were able to move into Turkey and (possibly) Persia, then that might have been a game changer. But even if the Russians had done that, trying to conquer and occupy those territories would have also taken up a lot of their time and resources. But they also apparently didn't like the idea of Russians in control of the Holy Land - a territory which has also had an enormous role in shaping the geopolitical perceptions of many people, including many in Britain and America. They may have been more concerned about that than any possible threat to India.