That is an excellent question.Can you be more explicit about the situation in China before going on?
To the best of my knowledge the OTL collapse of the Qing was a result largely of the belief that the regime had failed China in the face of foreign encroachment. Its military success was ensured to a degree by the Beiyang army - composed of more educated and liberal-minded recruits - largely going over to the republicans.
Certainly this is not an area of specialty for me, but I see a Qing dynasty that has failed more thoroughly than it did historically. So why is it putting up a fight more successfully than in our TL? I thought the Beiyang were dispersed and disloyal; what places them in the north and makes them loyal?
The Qing Empire has, to a considerable degree, failed worse than it did historically. This - along with Japanese support - is one of the reasons that the rebellions succeeded in ways they failed to do historically. The Republic has at its disposal much more manpower - and trained manpower - than it possessed historically.
Historically, the Qing collapsed not because of the Xinhai Revolution qua the Xinhai Revolution, but because of Yuan Shikai's decision not to back the Qing regime with the Beiyang Army, which was loyal (to him). But here he has no chance to link up with the revolutionaries, who are more confident that they can succeed without him. So he sticks with the Qing instead.
A distinction here is necessary: the Qing did possess other "modernized" army units other than the Beiyang Army (collectively and inventively referred to as the xinjun, or New Army) but these were inferior in number as well as (most egregiously) in quality. In the OTL Wuchang uprising, modernized units did ally with the Tongmenghui and the other societies that staged rebellions over the railway crisis, but these were almost entirely non-Beiyang elements of the New Army. (Frequently, accounts of the Wuchang events are garbled to make it sound as though the Beiyang Army itself comprised the entirety of the New Army and that it was elements of the Beiyang Army that spearheaded the uprising, which may account for your confusion.) The Beiyang Army by and large remained cohesive throughout the events of the Xinhai Revolution. In OTL, it was only after Yuan began to toy with making himself emperor and later, when Yuan died, that the Beiyang Army fragmented into disparate warlord cliques.
Back to TTL. Initially - as in, "in the first few months of the rebellion" - the north-south split was much less stark, but by the time Duan's army had reached Huainan, the republic and empire had both managed to more or less marginalize the major disturbances in their respective spheres of local dominance. Isolated elements of the Beiyang Army were mopped up (and sometimes enrolled into the Army of National Revolution) by Republican and Japanese "volunteers" around Wuhan, for instance, while uprisings in Beijing itself and in Shanxi were eradicated by Qing and Russian forces.