Plus, both the Austrians and Prussions/North Germans have potential friends elsewhere in Europe that presumably would not be unhappy to discomfit the British.
Best,
THis! and more.
In a ontinental war the brits simply don't have the manpower to put pressure on two of the most important powers in Europe. In wars against European powers the Brits always had strong allies or were able to strike at their enemies colonies - nothing is true here. Even if tehy manage to bring the French in (interesting possibility) all they would do is starting a longer war and be defeated in the end.
If we assume France stays peaceful and gets concessions from "Germany" (say Luxembourg goes to France

)
Major difference - Russia, Prussia and Austria grow closer (Britain is the common enemy for the next decades/France sits on the fence)
minor things (OTLs dars might be sped up/delayed here)
Mexican intervention butterflied away - or goes in the French way as Britain likel is NOT backing out because they want France to be an ally)
1866 war - Austria and Prussia are forced to work together
1867 - Maximilan NOT killed - was the more intelligent Habsburg and might be a good "face" of the Empire.
1870/71 - never mind
1878 - Russia (and the German states) carve up much more of the OE
1880+ scramble for Africa might see some different outcomes. (France might not back down at Fashoda (like incident) as it can rely on "Germanic" support).
All considered Britain can't win in the intervention.