Even if there is no Hitler, the UK will have to watch a rearmed Germany closely. I suspect Japan will expand at Britain's expense, though not as fair as Australia.
I think a higher-population France (if possible: I'm interested to see data or a plausible reason to show that an increase in population is plausible) may indeed "win" the Franco-Prussian War (massive butterflies aside), if only because all of those armies called up by Gambetta et al are going...
Lack of siege equipment would seem to be a fundamental problem. It took years for the Romans (and all their siege skill) to take Alexandria...where are the natives going to get this expertise? Outside aid? If so, you need to find someone to provide it and reasons for them to do so, etc.
Rule of thumb from OTL: a fleet loses 10% of its fighting efficiency for every 1000 miles it travels.
The German fleet must travel ~5000 miles to get to Puerto Rico from Wilhelmshaven, and over 2200 to get there from the Azores.
Assuming cruise speed, the German squadron will need to coal at...
All extrapolations were done by me. Anything unmarked with an asterisk, well, you provide the source for!
I don't really care; the point was subsidiary as a counter to any possible quibbling about the scenario I discussed.
Frankly, debating this is rather tiring and I'll let my points and...
Additionally, an 1870/71 partition of Austria-Hungary presupposes that the guy who refused to take the comparatively unimportant Sudetenland in 1866 (Bismarck) changes his mind four years later and annexes the predominant economic and population centers of another Great Power.
I took too long with my edit, I see. See my last post for a time-wasting analysis.
EDIT: As for possible rebuttals, a few more points:
1. If Austria-Hungary is in the war, Germany obviously doesn't keep so many troops in France, duh.
Moltke's war plan in the event of war with France and...
How you get, with an 1866 POD, the idea that an Austria beaten down (in tremendously unlikely fashion) by Italy would be more likely to enter war against Prussia than in OTL (i.e., they didn't at all) in 1870, while ignoring the fact that Napoleon III would clearly be more cautious in dealing...