Why did the French and British help the Qing instead of the Christian Heavenly Kingdom?

N3a is wrong re: opium; even if the Taiping had laws against it, the Taiping were buying just as much opium as the imperials were, so the opium dealers didn't really care either way.

Initially, the British were very careful to remain neutral in the civil war, with perhaps some favor towards the Taiping; they had no love of the Qing either, and the nationalism in vogue in Europe at the time generated some sympathy for Han Chinese rising up to overthrow the venal tartar oppressors (who had been giving them so much trouble over trade policy). They were aware of the Taiping's friendliness towards the West and their eagerness for modernization as well. Furthermore, from 1856-1860, the Westerners were in fact at war with the Qing Dynasty, but didn't consider this a violation of neutrality re: the Taiping Civil War.

While Hong Rengan had previously got his name out among the Hong Kong missionary community, the Taiping had difficulty establishing diplomatic relations with the Western powers; in Shanghai, they were in Imperial controlled territory. If they tried sending a diplomat to Shanghai, they'd have been executed out of hand. Furthermore, when they tried sending letters in 1860, Frederick Bruce (brother of the Baron Elgin who burned the Summer Palace), who more or less ran the show in British Shanghai, refused to open them; according to him, communicating with the rebels would be a violation of British neutrality, and furthermore, he had no faith in the Taiping whatsoever. Either their attempts to reach out to the westerners were a ruse, or if they were sincere, would alienate them from the other Chinese and make it impossible for them to rule China. He had no understanding of the Mandate of Heaven, and didn't know that rather than a disaster, rebellion was one of the elemental forces of Chinese political history. Because of this, he made up his mind that the Taiping could not be allowed to win.

Furthermore, the aforementioned letters contained vital information; the coming Taiping army under Li Xiucheng sought only to conquer the Qing held quarter of the city, not the American, French, or British districts, but, not knowing this, the foreigners blasted the approaching Taiping with heavy artillery. The rebels didn't even return fire; they had been assured that the foreigners had been told they had no quarrel with them. This, coupled with appalling ill discipline on the part of French troops involved in suburban fighting, actually generated more sympathy for the Taiping back in London and America, and in 1861 Henry Parkes had managed to hammer out an agreement with the Taiping, requiring them to stay 30 miles outside of Shanghai for the rest of the year, and allowing them unrestricted trade between the newly opened treaty ports in the Yangtze interior. This infuriated Zeng Guofan, who found a city he was starving out freshly resupplied by British food shipments.

So what happened? Well, the U.S. Civil War broke out in summer 1861. To the British, America, England, and China were deeply connected in global trade. Cotton grown in the U.S. south would be processed in north England and sold in China, and green tea grown in China was sold in the U.S. South. When Lincoln ordered a blockade of southern ports, the British could no longer buy raw materials from the south, nor sell consumer goods there. The 'cotton famine' as it was called drove up the price of textiles to the point that the Chinese wouldn't buy them, and the inability to sell the 2/3rds of the green tea they'd earmarked for sale in the U.S. south forced the British to dump their tea stocks on the domestic market, which led to a price crash. Britain had to restore order in one of their primary markets; they hoped throwing their support behind one of the combatants in the Chinese civil war would stabilize the market there, and get them greater trade concessions from the winner.

There's still the question of why they chose to throw their support behind the Qing, though. For one, the British actually have an embassy in Beijing after the conclusion of the Arrow War, which makes diplomatic communication with the Qing even easier. This is further compounded by the fact that the closest the westerners were to the warzone, in Shanghai, was in Imperial territory, meaning that communication from the Taiping to the westerners is impeded. It also took a long time for communications to make it back to the men on the ground in China; sending a message back to London took four months, and even worse, the MPs in charge of making decisions got their information from reports printed in the Blue Books. Thomas Taylor Meadows was an expert on Chinese history, but his letter forcefully defending the Taiping as the successors to the Qing wasn't printed until a year after it reached Britain, in 1862, by which point it was far too late.

Past the problem of communications, there was the question of religion. Here, the Taiping fall between two stools. On the one hand, even after Hong Rengan reintroduces Confucius to the Taiping examinations, the rebels have forever alienated the native Chinese scholar class, whose legitimacy was gravely threatened by the alien doctrines and whose wealth would be devoured by the Taiping's land reform program. On the other hand, the uh peculiarities of the Taiping religion made them blasphemers in the eyes of the Westerners. Keeping a harem of concubines, proclaiming to be the younger brother of Jesus, and receiving visions from the Heavenly Father all marked the Taiping as deeply idiosyncratic at best and outright satanic at worst.

Another fear for the British was the total collapse of order in China. They had no love of the Qing, but they feared that if the Qing fell, no state would replace it, and it would become another India. It would be Britain's 'moral duty' to conquer and colonize the entire country to restore order, and nobody wanted that. What they wanted was a government that would shoulder the costs of actually ruling China, while letting them reap all the rewards of unrestricted trade.

The crucial step for Britain to make war on the Taiping comes when Ningbo, one of the treaty ports the British had won in the Opium War, was threatened by the rebels. After taking Hangzhou, Ningbo was the obvious next target, so Frederick Bruce sent Admiral Hope's flotilla south to Ningbo as a show of strength; he hoped this would dissuade the rebels from attacking without necessitating any exchange of fire. Hope then dispatched a Captain Dew to remind the Taiping of their butchered attack on Shanghai two years ago and to aid the Qing officials in setting up defenses of the city. He further stretched the policy of neutrality by telling Dew to draw up estimates for how many European soldiers would be needed to hold the city. Furthermore, Bruce reckoned that since customs from the treaty ports were being used to pay the indemnity the British had imposed upon the Qing, then Britain had a right to defend all the treaty ports without technically violating neutrality, since the customs revenue constituted a British interest. This attempt to drum up war was undercut somewhat when the rebels enthusiastically agreed to grant the foreigners of Ningbo all the rights they'd enjoyed under the Qing, and the garrison fled rather than fight the assaulting Taiping army; there was no widespread violence when they conquered the city. Not that the local consular officer, Harvey, noted any of it, and he definitely didn't contrast the tranquility of the Taiping conquest with the massacre that had followed Zeng Guofan's siege of Anqing upriver, and further declined to mention the Taiping's offer to let the Manchu population of the city go peacefully.

Furthermore, Hong Rengan, who had previously held final command of all Taiping goverment officials and military forces, had found his power reduced when he left the capital in Nanjing, and his cousin had allowed the lesser kings to do as they wished. With the year of nonaggression promised to the British coming to an end, and Taiping control of the inner Yangtze cut off by Zeng Guofan's reconquest of Anqing, Li Xiucheng had but one option and one goal; the conquest of Shanghai. Without Hong Rengan to hold him back, the Loyal King gathered every cannon, every rifle, every grain of powder he could get his hands on, gaze set on the great prize at the mouth of the river.

As reports began streaming in to Shanghai of a massive rebel force gathering on the outskirts of the 30 miles radius, the newly reorganized Ever Victorious Army (an emphatically Imperial force, despite the foreign origins of its commanders) won a minor battle outside the city at Songjiang while an unexpected snow flurry allowed the foreign quarter to prepare for the real storm brewing on the horizon. After signing an agreement with the French for the defense of Shanghai, Admiral Hope worked out an alliance with the Ever Victorious Army's renegade commander, Frederick Townsend Ward. Ward's army would defend Shanghai, and Hope's gunships would support Ward's attempts to clear the Suzhou-Shanghai axis.

The actual inciting incident was almost laughably arbitrary; a Taiping commander who'd come back from a promotion loosed some celebratory gunfire, which had landed in the foreign quarter of Ningbo. Hope leaped at the chance, sending Captain Dew down to Ningbo, where after initially accepting the Taiping commander's apologies, received new orders from Lord Russel, which stated that Ningbo had to be kept in British hands. Dew drew up some impossible demands for the Taiping; they could no longer have cannons facing east, over the most likely avenue of attack, and from which direction he gave safe passage for a ragtag band of men and women armed with bamboo spears and bludgeons, and told him that repelling this force would be a violation of British neutrality. He got his war.