I think you've been misled by the rather inaccurate wording used in the history of these regiments who are keen to trace their origins back as far as possible.
In fact:
5th Royal Irish Lancers - Did not become lancers until 1861, they claim to trace their origins back to Wynnes Dragoons 1715.
9th Queens Royal Lancers - Also claim to have been Wynne's Dragoons but did not become lancers until 1816.
12th Royal Lancers - claim they are descended from the 12th Dragoons but were not armed with lances until 1816.
16th The Queens Lancers - claim they are descended from the 16th Light Dragoons, but were not armed with lances until 1816.
17th Lancers (The Duke of Cambridge's Own) - claim they are descended from the 18th Light Dragoons (Hale's Light Horse), but did not receive lances until 1822.
21st Lancers (Empress of India's) - claim they are descended from an East India Company Regiment called the 3rd Bengal European Light Cavalry, they did not become a lancer regiment until 1897.
So, as you can see the earliest appearance of lancers in the British Army was in 1816, beyond the timeline of ETW, although there might have been some sepoy lancers employed by 'The Company' in India prior to that date.
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