Of course I realize that the sepoys of the late 18th-early 19th centuries were soldiers in a private army of the East India Company. But the EIC seems to have made use of "regular" British units on occasion, and at the height of Napoleon's conquests and with fears of a cross-Channel invasion, it seems (at least to a novice like me) like a reasonable thought to do the reverse and bring in some of this large, well-trained army for defense. It's certainly something that the British Empire was willing to do in the World Wars, when India was directly governed by the British.
Was this ever considered? Or was it completely impractical for the logistics of the time? Or was there a fear of hosting basically a private army on or close to British soil that outweighed fears of Napoleon?
EIC Indian soldiers did serve on foreign (i.e., outside India) expeditions. The earliest such deployment I know of was 1762-1767, when 2,0000 sepoys were sent to Manila to fight the Spanish (16 KIA, and about 700 disease/starvation deaths).
Other foreign deployments were in Sumatra (1779), Ceylon (1781), Manila, Ceylon, Mauritius, South Africa (1794-1797 for these last 4), Egypt (1801-1802, 2901 sepoys fought alongside British and Ottoman troops), Java, Mauritius (1810-1811), Burma (1824-1825), China (1842-1860), Persia (1856), Ethiopia (1867-1868), Malaya (1875), Malta (1878), Egypt (1882), Sudan (1885), Sudan (1896-1897), China (1899-1900), Transcaspia (1918-1919) and multiple theatres in both WWI and WWII. Indian troops also served in the garrison of Saint Helena, which was under EIC control from 1658 to 1815.
The deployments in 1794 to 1797 were part of the French Revolutionary Wars - Mauritius was a French possession, the Philippines became enemy territory when Spain went to war against Britain in 1796, and the fighting in South Africa and Ceylon was part of the British/"Free Dutch" attempts to take Dutch colonial possessions are the French conquest of the Netherlands and the establishment of the pro-French Republic of Batavia as a Dutch puppet state.
The deployments to Egypt (1801-1802) and Java and Mauritius (1810-1811) were part of the Napoleonic Wars. Indian troops were sent to Egypt so that British forces could be kept in Britain to guard against invasion; the British troops who fought in the operation came from Turkey. This is the closest to what you asked about (Indian troops for the defence of Britain during the Napoleonic Wars); the Indian force indirectly contributed to the defence of Britain through allowing British troops to stay in Britain. The invasions of Java and Mauritius were carried out by British and Indian troops; India was well-placed geographically for the EIC and Indian troops to contribute.
The first European deployment appears to have been Malta in 1878 (which, reportedly, came as a shock to Britain's enemies, who expected that Britain would need to send troops from Britain to keep order in India if there was a European war). The first deployments in mainland Europe were in WWI, when Indian troops fought on the Western Front. In WWI, they also fought in East Africa, the Middle East (Mesopotamia, Egypt, Palestine), Gallipoli, Cameroon, China (whey they fought alongside British and Japanese troops against the Germans; the British motivation appears to have mostly been to keep an eye on, and restrain if needed, the Japanese troops), and Singapore (where they were sent to free British troops to be sent to France; there was fighting when half the force mutinied). In WWII, Indian troops fought in Europe, North Africa, the Middle East, East Africa, SE Asia, and the India-Burma theatre.
Further reading:
On foreign service of EIC Indian troops: Bandyopadhyay, P. K. (1990), "The Role of the Indian Sepoys in the British Imperial Wars Outside India, 1762-1801: Apportionment of Costs between the East India Company and the Imperial Government", Proceedings of the Indian History Congress 51, 706-713. http://www.jstor.org/stable/44148319
On reactions to the deployment of Indian troops in Malta in 1878: G. I. Wolseley, "The Native Army of India", The North American Review 127, 132-156 (1878). https://www.jstor.org/stable/25100657