Short answers, pulled from the story.
The Bengal War, also called the Anglo-Bengal War or the second Anglo-Mughal war, was a conflict between the British East India Company and a coalition of the Mughal Empire, the Bengal Subah, and the Awadh Subah. It ran from 1763 to 1765 and ended with a British victory sealed by the Treaty of Allahabad on the 16th of August 1765.
The Bengal War was triggered by a trade dispute between Nawab Mir Qasim and the British East India Company over customs duties on the Hooghly river. Company officials had created a system allowing goods with European passes to travel duty-free, devastating Bengali merchants. When Mir Qasim tried to reform the system and the Company refused, the conflict turned military in June 1763.
The Battle of Buxar was fought on the 23rd of October 1764, west of Patna, between British forces under Major Hector Munro and a coalition of Mughal, Bengali, and Awadhi troops. Despite being numerically inferior, Munro's forces won decisively, ending the coalition's ability to continue the war. Buxar was the decisive engagement of the Bengal War and led directly to the Treaty of Allahabad.
Mir Qasim was the Nawab of Bengal from 1760 and the central Indian leader of the Bengal War. He went to war with the British East India Company after it rejected his attempts to equalize customs duties between European and Indian traders. After defeats at Katwa, Udaynala, and Patna, he allied with Mughal Emperor Shah Alam II and Awadhi Nawab Shuja-ud-Daula, but the coalition was defeated at Buxar in October 1764 and Mir Qasim sued for peace in May 1765.
The Bengal War ended with the Treaty of Allahabad on the 16th of August 1765, granting the British East India Company control of Bengal's revenues through the diwani arrangement. Company officials treated the grant as a proto-constitution for British authority in the region. In 1773, the Regulating Act gave the Company complete control of the former Mughal province, marking the formal beginning of direct British rule in Bengal.
The Company secured a lease on the river Hughli in 1658 and built Fort William there in 1696, establishing a lasting physical presence. In 1717, Mughal Emperor Farrukhsiyar issued a Farman granting the Company trade rights in Bengal; Company officials interpreted this as conferring broad sovereignty over the province, placing them on equal footing with the Nawab himself. Bengal's drift into near-independence from the Mughal center after Aurangzeb's death left no unified authority capable of imposing limits on the Company's growing influence.