Have the Napoleonic Wars end in a draw.
That's how back you'll need to go for it to happen I think. And even that's a stretch.
With Napoleon III at the helm, France WILL NOT fight Britain, as one of N3's main foreign policy planks was rapprochement and alliance WITH Britain.
But any ways, in the ASB scenario that history and the geopolitical situation in Europe relatively closely as OTL, the Union still wins if it's in anywhere near the shape it should be (because the Union significantly dwarfs the capacity of any major European power to project on the North American continent by the 1860s: I'd recommend TFSmith121's
Burnished Rows of Steel TL for an understanding of that), the Mexican adventure ends for Napoleon III (or never occurred) and France loses her most far-flung colonies, eg. in the Pacific, and...France and Britain stare daggers at each other over the channel. Prussia and Austria are too busy preparing for a fight over the supremacy of Germany; Italy had recently gained independence in no small part to the assistance of France, while Russia is looking inwards after the Crimean War (and is otherwise embroiled in The Great Game with Britain), so there is no continental interest in fighting France at the time.
Plus the British Empire is markedly stretched in its army and naval deployments maintaining the empire, so...nothing really happens.
Again, France and Britain fighting it out in the 1860s is massively ASB unless you go REALLY far back. To a point where the ACW is probably not recognizable.