Since OP is asking for earliest possible date, here are some dates, going backward in time.
1861: the Tongzhi Restoration happens, but Prince Gong and the reformists manage to purge Cixi afterward instead of the other way around, leading to Tongzhi being competently raised by pro-reformists instead of being deliberately spoiled by Cixi to be her pawn, and China goes forward with an emperor intent on modernization.
1850: Prince Gong becomes the next emperor instead of his half-brother. Prince Gong leaned pro-modernization (that is, he placed it as a bigger priority than the fiscal issues), while Xianfeng was more like his father. So if Taiping still happens, he'd be more favorable to going forward with modernization as quickly as possible. Also, Prince Gong lived much longer IOTL than Xianfeng, and Cixi was not his concubine, so even if he died earlier than IOTL, she would not have control over his succession.
1850: the Taiping Rebellion doesn't happen, the silver crisis passes naturally in the mid-50s, and Xianfeng can now consider modernization.
1813: the BEIC's monopoly on British trade with China is revoked, allowing British merchants to flood opium into China, which causes the Opium War to come early.
1810s: the Mexican revolutionaries are suppressed, and the silver crisis is prevented, for reasons outlined in a previous post.
1805: the British capture the Philippines off of Spain in the War of the 3rd Coalition. This gives them a foothold in the South China Sea and increases the possibility of an early war between China and Britain, regardless of opium. The British wanted free trade since the failed Macartney Embassy, and they might be more interested in being assertive about this if they have a fleet base next to China.
1790s: Napoleon doesn't come to power, France doesn't (successfully) invade Spain, Spanish authority in Mexico doesn't break down, the Mexican War of Independence is prevented or delayed, and the silver crisis is prevented.
1793: the political fallout of
the Hastings trial is worse, which causes Parliament to revoke the BEIC's monopoly on trade with China in the 1793 charter.
1770s: the British avoid or suppress the American Revolution. IOTL, American merchants supplied about 10% of Chinese opium imports by the 1820s. ITTL, they'll add to the number of British merchants clamoring for the BEIC's monopolies to be revoked, which would lead to the China monopoly being revoked in 1813, or possibly even in 1793 if they use the Hastings trial to attack the BEIC. Also, if France doesn't support the Americans and focuses on their own fiscal problems, they could avoid the French Revolution. This prevents the rise of Napoleon, the invasion of Spain, and the collapse of the Spanish Empire in the 1820s, which prevents the silver crisis.
1763: the British forces which
captured the Philippines in the Seven Years' War manage to get news of their victory to London before the Treaty of Paris is signed, and Britain annexes the Philippines. This gives Britain a foothold in the South China Sea, etc, etc, except 40 years earlier. 40 years is a lot of time in which they can instigate a war which shakes the Qing government into a modernization mindset, especially after Heshen is purged. Then again, Heshen only began to work in the imperial palace in 1772, so this could easily avoid Heshen's era entirely.
Also, there's a major what-if that would strongly influence the situation, but would probably have to be paired with other events to significantly change the situation.
1824: the Burmese press onward to the undefended Chittagong after the Battle of Ramu instead of retreating, ravage Bengal, and ultimately defeat the BEIC. The BEIC, having incurred massive debts in the war and lost its prestige, defaults on its debts and is disbanded by the British government. With its monopoly on opium trade gone, independent British merchants flood into China with opium, causing the opium crisis to happen slightly early. From there, who knows what happens?
