1868: King Wilhelm IV of Prussia (OTL Kaiser Wilhelm I of Germany) dies of pneumonia, aged 71; succeeded by his son Friedrich III.
1870: Bismarck attempts to engineer war with France, without bothering to consult King Friedrich, and is sacked.
1871-1876: Friedrich pursues German unification, in alliance with German liberals, but faces obstruction by France.
1877: Grand Duke Frederick I of Baden has an idea: win French support for German unification by offering a dynastic upgrade to the House of Bonaparte; to wit, marriage with the House of Hohenzollern. Friedrich's eldest daughter, Charlotte (b. 1860), will marry Napoleon III's son (b. 1856). In return, France will back unification and Friedrich's elevation to the rank of Emperor. The two young people go along, and the marriage is reasonably successful (three sons, two daughters). Even Britain supports the alliance, as Charlotte is Queen Victoria's granddaughter, and will now rank with her as Empress. (A famous 1888 photographic portrait of Victoria, her daughter Kaiserin Victoria, and grand-daughter Impératrice Charlotte is titled "Three Empresses".)
Everybody gets along fine for many years.