I'd wager tens of millions directly, hundreds of millions indirectly. The tougher aspect of that is calculating what number of those would have died anyway and which would have survived had the British Empire not intervened - and substracting the ones who survived thanks to British intervention on the rare occassion when they would have died without it.
The famines in India are particularly difficult. There's no doubt that the colonial administration's attitude worsened some famines. At the same time, India has always suffered famines. Take this one in 1791-1792 which primarily hit pre-colonial parts of India:
https://en.m.wikipedia.org/wiki/Doji_bara_famine
Upwards of 11 million dead, caused by an El Nino. Colonial regions were actually hurt less severely, probably because they had access to imported grain to a higher degree. And in an interconnected world, who gets which share of the blame for the Bengal famine? It was triggered by the Japanese invasion of Burma, after all, and exacerbated by Nazi German military pressure on the UK which reduced the British ability to respond.
There's nothing uniquely evil about the British Empire. It behaved the exact same way as the other colonial powers. Not to mention that it existed during a period of history where truly benign rulers were very few and far between, with the states of the non-colonised world typically consisting of feudal or monarchic governments. I'd argue that tribal populations likely experienced the most severe repercussions of colonisation because they were, after all, self-governing, but I'm not sure if being a serf under a Mughal Emperor was any better than being a peasant subject of the Queen.
It is hard to even count. You can trace the current conflict in Syria back to the British Empire, for example. Even the idea of dominant minority rule.
The spread of capitalism and exploitation through the East India Company, borders all over the world drawn on a desk in London without knowledge or respect for the local circumstances...how could you ever put a number on that?
Yeah, no, the British only held Syria for about five years during WW2. They had basically nothing to do with the country. Before that it was ruled by France for some 20 years, but the greatest influence bar none before independence was obviously the Ottoman Empire which ruled the country for over 400 years. Dominant minority rule is hardly a British invention either. Case in point being the Mongol Empire, Roman Empire, Mughal Kingdoms, or any other time in history where one people conquers and rules another.