Bismarck released his carefully worded readout in the evening of Thursday, 14 July 1870. Assuming the french were "making the bridge" for Bastille Day, they would have had one working day (Monday, 18 July 1870) to fact check, but I don't know if they took even that, because according to https://en.wikipedia.org/wiki/Causes_of_the_Franco-Prussian_... , on 19 July 1870 "Le Sourd, the French Chargé d'Affaires, delivered Napoleon [III]'s declaration of war at the Foreign Office" in Berlin.
https://scholar.dominican.edu/cgi/viewcontent.cgi?article=11... suggests the french were already mobilising on 14 July itself, and that while telegraph use may technically have been possible, US diplomats were still communicating by snail mail with State.
That emotion won over fact checking is supported by Ch. I p.3
> Washburne dismissed the accuracy of the Ems telegram because it had been challenged by King Wilhelm. "There is no truth in the reports concerning the indignity which the King of the North German Confederation offered to Benedetti, the Envoy of France." Washburne based this statement on a published denial by the King which he had read in Cologne while returning to Paris. Though he questioned the veracity of the Ems telegram, Washburne still credited it with arousing the French Chamber of Deputies to vote for war.
(the war vote was Friday, 15 July. It might be interesting to see if any french papers had carried Wilhelm's version of events.)
Depends how tense the situation already was. A well-timed tweet could be enough to tip things over the edge if both sides were already on the edge, and I don't think we're too far away from that.
As for the blackmail and war starting you probably couldn't do much with public tweets (nobody is going to go to war over a tweet without fact checking it) but access to private messages is an entirely different story.