An interestingly turbulent time, one that's only had two oblique references in the Whoniverse. Lots of opportunities for a party of out-of-place travellers o get into trouble....
Loosely I'd group possible scenario into four categories:
1. Historical A rarity in Who these days but it'd be quite possible, and perhaps a refreshing change, to drop the PCs into a situation that's free of alien or other weird influences. There are plenty of problems for them There was a lot of violence during the uprising, probably more than a hundred thousand dead overall. The 'Boxers' attacked Westerners, Christianised Chinese and anyone who didn't 'look right'. So a party of time travellers could easily be separated from their transport, and each other, and scattered into hiding. Or stuck in Peking in the Legation quarter during the siege. What do they do? Do they scrupulously avoid meddling? Or improvise a few things to assist? Do they try and stop the violence of both sides? And it's also a good place to inflict cholera on your players.
2. Almost historical My term for a historical setting with some oddness, unconnected to events, but no major outré elements. Perhaps on arrival the party accidentally alter some crucial detail and put the river of history onto a new path. Or they discover something, unconnected to the uprising, that's clearly out of place and investigate, while avoiding the mobs, the Eight-Power troops and looters.
3. Pseudo-historical The classic Who adventure; something weird is happening, maybe it caused the uprising, and needs to be dealt with. Perhaps the players needs to ally with the Boxers and/or the foreign troops? Or maybe someone is using the cover of the chaos for their own purposes; collecting slaves to brainwash to operate her secret inter-dimensional base perhaps? Extracting interesting chemicals from humans? Looting art works? The latter has the potential to introduce other time travellers, such as the Alexandrian Society who'd find plenty of opportunities for acquiring artworks.
A note on looting. It is one of the unwritten laws of war that a city which does not surrender at the last and is taken by storm is looted. Lieutenant Colonel Henry Vaughan.
This happened a lot during the uprising and especially afterwards. The First Secretary to the US legation, Herbert Squiers, a noted collector of porcelain, notoriously filled several railway cars with loot and artifacts. Every afternoon, except Sunday of course, the British Legation held auctions of looted goods outside the main gate, though "Looting on the part of British troops was carried out in the most orderly manner". These continued well into 1901. Another active market was the Catholic North Cathedral, described as a "salesroom for stolen property".
Or maybe they were searching for something more unusual? Alien artefacts perhaps? Lost time machine (such as Greel's cabinet) to extend their capabilities perhaps?
4. Meddling Finally maybe someone wants to alter events significantly. While actually united China would require serious alterations to history even earlier maybe the party notices such changes during the uprising and travels back to (say) the late 1860s to prevent them. Mortimus is an interesting candidate, perhaps disguised to appear Chinese. Or possibly a group of Chinese Nationalists for a future period who seeks to save their country from decades of strife.
Some other points.
1. Greel. While Greel is dead (supposedly) by 1900 and the Time Cabinet in London, perhaps another escapee from the fifty-first century followed a similar course and arrived a little later. Possibly arriving in the midst of a group of Boxers and being mistaken for an indication of divine support for their cause? Assuming they also require periodic life force infusions there'll soon be a train of desiccated corpses for teh PCs to stumble on, and be accused of causing.
2. First Doctor and Susan Were involved in the Boxer Rebellion, according to a comment in The Fire of Cadiz. He used smoke bombs to escape. So there'll be at least one other time traveller there
3. Jack Harkness. Was also present, doing something with explosives. Demolitions (for whom)? Safecracking?
4. Torchwood. Might also be around. Perhaps they've learned about Greel's Time Cabinet and are tracking back for find other anachronistic artefacts?
5. Cholera A fun disease to inflict on players (and a useful excuse when a player can't make a session). The Sixth Cholera Pandemic was happening (killing about 600,000) but didn't effect China that much.
6. Fu Manchu. Cay Van Ash speculated that Doctor Fu Manchu had been a member of the imperial family of China who'd backed the Boxers, an hence gone into exile and into fame. I've speculated elsewhere about a Doctor Who Manchu hybrid campaign with elements of The Talons of Weng Chiang. The events of the uprising could be a prequel.
7. 1910 Plague Topical... A bit later but the Manchurian Plague of 1910-11 killed at least sixty thousand people and was unusual for being almost entirely the pneumonic form of y. pestis, as opposed to the usually more common bubonic from. While this might have been a biological test by Fu Manchu, perhaps aimed at the Japanese and Russian encroachment, it might be linked to earlier events. Perhaps it wasn't actually plague?
8. War Lords The War Lords scooped troops from this period. Were they returned without memories of those events? Might a few people from other time periods have been mixed in? With anachronistic weapons and amnesia... Or was there a War Lord scouting party there, complete with SIDRAT and armed guards, scouting the period?
9. Mind Parasites While Sympathy for the Devil is explicitly an alternate universe the Master stated that the mind parasites he used in England and China came to Earth in the late nineteenth sentry and caused the Boxer Rebellion.
10. Disaster tourists I've also speculated about "disaster tourism" previously (specifically the films Thrill Seekers/The Time Shifters and Timescape/Grand Tour: Disaster in Time). Might the Boxer Rebellion be on their schedule? It's not a natural event and presumably some security would be needed, but that increases the opportunities for meddling, whether deliberate or accidental.
Hope this helps.
Time, time,time, see what’s become of me. While I looked around for my possibilities... My AITAS files.