This is a longshot, but doable.
Bismarck retains power until his death, and the Russian alliance survives until then. (Russia, despite the credit crunch, was not eager for the French alliance, and was rather happy to be led about by Germany, despite significant opposition from the Francophilles.) Otto Pflanz makes a convincing argument in his magisterial biography of Bismarck that economic tensions did not have to lead to a break with Russia diplomatically, while there are diplomatic historians who argue that Russia entered into the alliance only because of diplomatic isolation resulting from the post-Bismarck break and because it misread Wilhelm II's clumsy initial Anglophil policies as wanting to create a Anglo-German alliance against Russia.
Bismarck and then his successor cosy up to France while France is in throe of Anglophobia (Fashoda was merely the culmination, and the French Right seriously thought about allying with Germany, both because of Anglophobia and because of the Dreyfald Affair). At this period, there were serious people in France who thought an entente with Germany was possible. One major French newspaper opined that while the Germans were regrettably their enemy at the time, it needed not be always thus so; in contrast, England was France's eternal enemy.
What this setup would do would be to actualize the continental league that Germany talked about (but wasn't serious) at the time of the Boer War. With Russia in German pocket, France would be tempted, and more importantly, the Russian angle would enable Germany to supply the Boers with arms and "volunteers." In RL, neither Germany nor France could do jack to help the Boers, even if they wanted to (and Germany didn't, though France seriously wanted to), because of the RN. With Russia in tow, the League could threaten India on land, a threat that was 100% BS but the British FO took seriously at the time.
What this might let happen would be a proxy war of the 20th century kind in S Africa, where the Continental League would give vital materiel and human resources for the Boers to continue their conventional warfare and deny Britain the ability to create concentration camps.
The best case scenario would be for the British public to tire of the drain in men and gold, and give pro-Boer MPs the leverage they needed to convince the government to come to terms with the Boers. The worst case scenario would be for the British to get fed up with the League and Copenhagen German/French/Russian fleets and start up a war.
The most likely scenario would be for the League to crack under mutually incompatible national interests in the face of British emnity, and the Boers would lose, but without the advantageous peace they got in RL, because the British public would be pissed off at them for the tremendous losses in blood and gold.