While British and French forces ultimately supported the Qing, did the Heavenly Kingdom have any vocal sympathizers in the West? If so, did any of these supporters attempt to support Taiping efforts directly (perhaps financially or directly militarily)? Was there sympathy for the Heavenly Kingdom in other parts of East Asia such as Japan or Vietnam?
As /u/voyeur324 said, I've written quite a few posts before about Japanese reception of the Taiping, Western policy towards the Taiping, Taiping foreign policy, and mercenaries and volunteers fighting for the Taiping and Qing. However, these generally concentrate (especially in the latter cases) on why there was Western opposition to the Taiping, so this is your lucky day because I'm now going to have to do a whole new post to address your specific question on pro-Taiping sentiments.
Broadly speaking, the foreigners (both Asian and Western) who interacted or engaged indirectly with the Taiping can be divided into three groups: state actors, missionaries, and private interests. As a rule of thumb – but not a hard-and-fast rule – state actors skewed anti-Taiping, missionaries were split, and private interests skewed pro-Taiping. I say 'skewed' here as there were a good handful of pro-Taiping state actors, and ditto for pro-Qing mercenaries and financiers. Let's look at the general trends first and then dig down.
It should be somewhat unsurprising that, by and large, state actors did not eagerly support the rebels. Even if the Qing were a foe, they were a familiar one, and Western powers who had already secured major trade concessions, especially under the terms of the Treaty of Tientsin (1858) and Convention of Peking (1860) would have little reason to want to back the Taiping when there was no guarantee that they could be coerced into inheriting those arrangements. The Japanese delegation on board the Senzaimaru in 1862, however, had slightly different motives for being anti-Taiping, namely delusional conspiracy-mongering about the rebels somehow being a Western plot to usurp China from within.
Nor should it surprise us that missionaries were uncertain what to make of the Taiping, whose religion was at once so familiar and yet so alien. The Taiping were, ostensibly, Christians. Their doctrine was Biblical, they held weekly communal gatherings involving hymns and prayers, they actively destroyed symbols of China's established religions. Yet at the same time, their leader professed to be the literal second son of God, their Christology was decidedly non-Nicene, let alone non-Chalcedonian, and the jury was still out on the validity of using of Chinese ritual practices if the end goal was still Christian.
Private individuals, however, especially merchants or people from mercantile backgrounds, saw in the Taiping fertile grounds for further commercial exploitation. In fact, more than a few of the aforementioned state actors shared this perception, at least in the years before the Arrow (Second Opium) War. While many merchants may have been relatively ambivalent on the ideological aspect of the Qing-Taiping conflict, there were those who backed the Taiping for pragmatic reasons – after all, remember Rule of Acquisition #34: 'War is good for business.' The smuggling of weapons and supplies into Taiping territory could be an incredibly lucrative business, and while those who actually plied this illicit trade have left little record themselves, there is a substantial amount of evidence to show they existed – more on this later.
So that's the general trends at work, let's go through this in a bit more depth chronologically.
From 1851 until 1856 (what I will term the 'early period' of the Taiping War), the general foreign attitude was pretty positive. As I note in the first of my linked post, the popular press in Japan seized on what seemed like a major anti-Manchu uprising, but also used the conflict as a conduit for fears of foreign imperialism in Asia, predicting that one side or another would receive decisive support from the Western powers. For missionaries, this seemingly Christian movement was a massive boon, with Protestants especialy hailing the apparent triumph for the true faith that the uprising represented. And, in this pre-Arrow War period, there were those in the service of foreign states who saw the potential for a stronger commercial partnership with the rebels than the imperials. From the latter group, the most significant was probably the American special commissioner Humphrey Marshall, who butted heads with Commodore Perry over the allocation of American naval resources in the western Pacific. 'There is nothing to be hoped for in Japan,' he declared, 'equal to the advantages now actually enjoyed in China.' Marshall was convinced that a Western-sponsored Taiping victory would bring vast benefits to the US, while three other diplomatic missions in 1853-4, two British and one French, also went in with a degree of optimism. However, the three countries failed to secure any significant agreements, and came out with more than a little bewilderment at the Taiping's fanaticism towards their 'heretical' religion. Still, more than a few were still quite optimistic. Joseph-Marie Yvan de Callery's L'Insurrection en Chine and Theodore Hamberg's The Visions of Hung Siu-tshuen, published in 1853 and 1854, respectively, were the first major publications on the Taiping in Western languages, and both generally supported the Taiping cause. On top of that, a number of Western mercenaries began working for the Taiping in various capacities. Only one from this early period, an anonymous Irishman whose testimony was printed in the Hong Kong newspaper Friend of China, survives, but what makes it really especially notable is that this mercenary was in Nanjing during the coups of 1856, making his one of the only eyewitness accounts of what transpired during and after Hong Xiuquan's assassination of his ambitious lieutenant, Yang Xiuqing.
The 'middle period' of 1856-60 was altogether quieter, but it is clear enough that there continued to be missionary visits, arms shipments from Shanghai, and guns for hire, erm, hiring themselves out. Western attentions were, at this time, naturally occupied mainly by the Arrow War, and in any case the coup period of 1856 had weakened the Taiping significantly and reduced their presence on the radar. There remained some servants of the state who were pro-Taiping: Thomas Taylor Meadows, an early Sinologist and the British consul at Shanghai until 1860, saw the Taiping as the latest expression of a cyclical process of dynastic decline and renewal and advocated British neutrality in order that the cycle could play out naturally. However, he was very much in the minority, with more aggressively anti-Taiping (or in some cases generally anti-Chinese) officials like Frederick Bruce and Harry Parkes being the norm.
The reall resurgence of interest in the Taiping came after their advance on Shanghai in the summer of 1860, shortly before the Anglo-French expedition to Beijing which concluded the Arrow War. The conclusion of the Arrow War is the key change here. After October 1860, no longer was it a choice between a potentially pliant Taiping regime and an intransigent Qing, but rather between a known unequal arrangement with the Qing and a complete unknown with the Taiping – a complete unknown that was threatening to nullify the concessions only recently seized from the Qing. Thus, despite the best efforts of individuals in the Taiping camp like Hong Rengan to court Western support, rational policy favoured a pro-Qing intervention over maintaining neutrality for much longer. Those who remained pro-Taiping through this period were distinctly in the minority, at least among state-affiliated actors. Nonetheless, the aforementioned smugglers were still very active. Augustus Lindley is not explicitly clear on when he arrived at Nanjing except that it must have been between late 1860 and early 1862, but this issue aside, his travelling to Nanjing on a smuggler carrying foreign currency implies ongoing regular Taiping-foreign contact. Lindley was but one of dozens if not hundreds of mercenaries who fought for the Taiping even after 1860: Consular official Robert Forrest, who later wrote quite glowing retrospectives on the Heavenly Kingdom, on one occasion arrested 26 of a group of 104 Western mercenaries in Taiping service. Lindley at one point held command of a group of around 25 Westerners based in Suzhou, the activities of which were briefly recorded by his successor, George Smith, from October to November 1863. What's rather intriguing is that Western volunteers continued to support the Taiping until quite late. At one stage, ex-Ever-Victorious Army commander Henry Burgevine defected to the Taiping with 70-odd Western officers for a few months. After the fall of Nanjing in 1864, a Taiping holdout under Li Shixian apparently received some British volunteers, including two of Charles Gordon's former officers. What induced them to defect and why they did so is unclear (money troubles, perhaps, owing to the disbandment of the Ever-Victorious Army a few months earlier?) but it still shows the existence of Taiping sympathisers quite late in the day.
/u/EnclavedMicrostate has written more about the Taiping on this subreddit than any other user. Here are some highlights: