South Asian here. During my history classes in school I was taught how the British gained power and conquered much of South Asia around the 1600s. However none of the sources that I read ever mentioned the British employing slavery in these lands. However at the same time slavery was widely practiced in American colonies like Barbados and others.
Either I am missing something, or the British really did not employ slaves in South Asia.
Can someone explain why this is so?
The British only slowly became active in South Asia in the 17th century. Up until then they were in competition with the Dutch and French. Only after the Battle of Plassey in 1757 did the British began expansion in India. This is the date that is taken to be the beginning of the colonial period.
As to why they did not import slaves, they did not have to, there were already local slaves and indentured labourer. For the most part the British did not overthrow local governments but rather co-opted them. Even after 1833 and the abolition of slavery in the rest of the British Empire, slavery in India was allowed to continued as it was seen as an "indigenous institution" see p.43ff.
Before and after the abolition of slavery in India and South Asia debt-bondage was used in place of slavery, and has in fact been referred to as "A New System of Slavery" in the literature. There were even Indian indentured labourers exported to the West Indies, in fact after the end of slavery indentured labourers were exported from India all over the Empire.