I can't speak to the American side of this, but I will discuss the First Anglo-Maratha War. The First Anglo-Maratha War was a bit of a farce, and quite humiliating for the British from start to end. Started by the Bombay presidency in support of Raghunath Rao, the leading claimant to the Peshwa-ship of the Marathas with whom they had signed the Treaty of Surat. This was all undertaken without permission from Warren Hastings, recently empowered as Governor-General by the regulating act of 1773, in Calcutta. The recent act left the British in a great deal of confusion about where authority lay, and it is very evident in the early stage of this war. The Bombay’s troops advanced on the Maratha’s and defeated them in several engagements when orders from Calcutta arrived to order a cessation of hostilities and negotiations to begin. The Marathas were also confused. The rival Peshwa was still a child, and the regency junta was divided among itself. In February 1776 Calcutta ordered Bombay to resume its attack, at the same time that word had arrived from the Maratha capital that negotiations had been completed, and from London that the original treaty was approved.

It is only really at this point that the American War of Independence begins to affect the course of the war in India, with the entry of France into the war. The French were interested in expanding the war into India, as they had during the seven years war. French agents were reported to be in the Maratha capital, which made the English anxious that French troops would soon arrive. The war was resumed, and reinforcements were sent marching across India from Calcutta to the aid of Bombay. The army would arrive only after the army of Bombay was defeated and forced to sign a humiliating treaty and slink back to Bombay, where the treaty was immediately torn up, and the war recommenced. Further reinforcements were sent from Calcutta by Hastings, which lead to a duel at dawn on August 17th, 1780 between the Governor-General of India and Philip Francis, a member of the supreme council of Bengal, where, after being taught how to use their pistols by their seconds, Hastings wounded Francis. These reinforcements managed to secure victory, and a final treaty was ratified by both parties in May 1782.

It is important to note that the reinforcements the brought the war to an end were dispatched from Bengal, not from England. There were British regiments in India, most notably the Highland regiments that made up a significant part of the army surrounded and destroyed by Mysore at the Battle of Poliur, but the East India Company’s armies own white and native regiments made up the majority of these armies. The wars in India would not have been a significant diversion of resources from the fighting in America and separated from Europe by six months of travel the course of the two wars could have very little relation to the other. As for why they are not mentioned together more often there are two reasons. One is before discussed independence. The other, the greater one is that none of the principal European forces involved has much reason to talk about it. The Americans were fighting for their independence on the other side of the world with the French, and the British were dealt several humiliating defeats by Indian armies that had effectively adopted European training and equipment, and acted quite dishonorably, squabbling among themselves and repeatedly breaking treaties. This period was not a bright time for the British in India, and has on either side of it are the triumphs they enjoyed under Robert Clive and Arthur Wellesley. Patriotic, jingoistic historians would be unlikely to discuss it unless absolutely necessary. I hope this answers your questions.

Sources

Dalrymple, William, and Olivia Fraser. The Anarchy: The Relentless Rise of the East India Company, 2019.

Heathcote, T. A. The Military in British India: The Development of British Land Forces in South Asia, 1600-1947, 2013.

Keay, John. The Honourable Company, 2017.

Munro, Innes. A Narrative of the Military Operations of the Coromandel Coast. Miami, Fl: HardPress, 2017.