There are a couple of references in "Plain Tales from The Hills".
One seems quite important:
but Tods knew the Red Lancer and most of the Members of Council personally
Others seem like regular guards:
Red Lancers came in, and the Head Chaprassi, who speaks English, came in, and mace-bearers came in, and ladies ran downstairs screaming “fire;”
The closest I can find is 16th The Queen's Lancers, they were known as "Scarlet Lancers" and stationed in India but they were a cavalry regiment.
Its difficult to say without seeing more of the context as I'm not familiar with Plain Tales from The Hills, but most British cavalry regiments wore blue rather than red. The 4th Bengal Lancers wore red though, so it may be referring to them. I have no idea if this is actually the case though.
Any Lancer would have to be in a cavalry unit (at the time, one armed with lances). In both references the context appears to be of guards on duty.
In the first reference, in the eyes of the child "Tods" the impressively red uniformed guard is as important as the Members of Council and the child is on speaking terms with all of them.
In the second reference the red uniformed guards appear when smoke envelopes the Viceroy's afternoon tea.
Although the tales were published in the 1880s Kipling doesn't identify when they were supposed to have happened (they are fictional). The 4th Bengal Lancers weren't named Lancers until 1900, and the 16th Queens Lancers were finished their India posting in 1846. Before 1840 when most were changed to blue, red was common for lancer uniforms, and of course it is possible that he is describing the soldiers as red lancers because the facings on their blue uniforms were red.