Suppose things go bad in the Fashoda Incident (that single nervous soldier...) and we get a war between Britain and France erupting in Africa in 1898. I think that Germany would be happy to jump on France's back to reap their own share of the glory while Italy sit out to see who wins and then jumps on the back of France.
True for Germany, but wholly wrong for Italy. Expecting Italy to behave as IOTL Great War is rather unreasonable. An European coalition with Britain and Germany shall always look definitely stronger than the opposition to Rome, they are never going to fight it. And they have little incentive to staty neutral, too. In the 1890s, their foreign policy and strategic stance is still strongly anti-French and pro-German. And France has a lot of irredentist and colonial claims that Italy covets: Nice, Savoy, Corsica, Tunisia, Dijbuti, eastern Algeria, the Riviera. Maybe to a somewhat lesser degree than the Austro-Hungarian claims, but they are not going to pick the definitely weaker coalition (they are not really going to care about America) for it. French stuff is more than good enough to justify a war, and they have been preparing to fight this one for a couple decades. They declare war to France and Russia a few days after Germany does.
Russia is a French ally and will have to assist and they'll likely attack in Central Asia as that is where their borders with Britain are while Germany finishes France first.
It depends on what the Triple Alliance decide to do, and to what degree they coordinate their strategy with Britain. This is pre-Plan Schliffen, after all, so France first is not a given. But I think that it is a reasonable assumption that, by relying on the 1870 experience, The CPs decide to finish France first and then wear Russia down gradually with encirclement. So defensive stance on the Eastern front, since in 1898 Russian mobilization shall be even slower than in 1914, and concentrating the bulk of Italo-German manpower against France (Italy activates standing protocols to send part of its forces on the German front, it may otherwise take an offensive or defensive stance on the Alps). Britain takes a defensive stance (initially) Central Asia, the BEF may either land in Belgium or make landings against Channel and Atlantic French ports. Germany and Britain are likely to send Belgium a "polite" request for military access. Belgian response is a toss-up: they might refuse and plead for French help, or concede transit to the CP: they are genuinely committed to neutrality, but defying their British patrons is terribly risky (London might decide that after all, resturning it to Netherlands is te best option).
The main issue here is that France, especially if Britain and Germany go through Belgium, and the latter concedes transit, is quite overstretched and has strong chances of failing to stabilize the front with trench warfare. There may be no "miracle of the Marna" here.
The Japanese will be wise to assist in the butchering of France and Russia and not oppose Britain so we'll get a Russo-Japanese war a few years earlier unless Tokyo remains neutral of course as Russo-Japanese sentiments weren't that bad yet (although that'd anger Britain). In the chaos of war, they might try another stab at China again since their gains were cut short after the First Sino-Japanese War (1894-'95).
There was already some Russo-Japanese antagonism building about Korea and Manchuria. Tokyo may either backstab Russia (say 65% chance) or China (say 35% chance) unless Russia concedes Korea and southern Manchuria without a fight.
The Ottomans will likely be wise enough to stay out of this slaughterfest unless Russia is dumb enough to attack them first and make it a three-front war for them.
True, but the wild cards here are the Balkan powers. At least some of them are going to exploit the chaos and try to grab a piece of the Ottoman or the Habsburg empires (and their rivals might cooperate, as they did in the First Balkan War, or take the opposite side, esp. Serbia and Bulgaria). If some Balkan states attack Turkey, and bit more than they can chew, Russia would be forced to expand the war or left its southern flank exposed anyway. Depending on diplomatic and political butterflies, you have various options here, from the Balkan states trying an Ottoman curbstomp, Greece staying neutral (fearing Anglo-Italian invasion) and Serbia + Bulgaria making an anti-Ottoman alliance of convenience, Serbia + Romania trying to backstab Austria, Serbia-Romania entering for Russia and Bulgaria for the CPs.
If anyone knows a plausible way to get the US in on either side (please tell me if you do), we have a three continent spanning world war on our hands. with these sides:
Your best chances are the Spanish-American War joining with the Fashoda War (Spain and France making a "Latin League" against the Anglo powers) or America deciding ot make a grab for Canada (say there was a skirmish about the Venezuelan crisis a few years before, this caused lingering Anglo-American antagonism, and America decides to try a comeback while Britain in busy in Europe and Central Asia).
France & Russia and possibly China (maybe the US over the old 'answering the call to Lafayette', the Open-Door policy in China or over good Russo-American ties or any combination of these although I think the British/Germans would have to do something incredibly stupid first to get them out of isolationism).
Those motivations are near-ASB for isolationist US to pick a fight with Britain and Germany, esp. the call to Lafayette, who bloody cares about that, apart from a tiny romantic fringe ? You need a casus belli that may be relevant to American interests in the New World. What about this: sub-PoD: Britain and Germany have a period of detente and stick together during the Venezuelan crisis, America is humiliated and does not forgive easily. During the Fashoda War there are some naval incidents between American ships and the Anglo-Germans, with loss of American lives.
I know France will lose, but I'm more interested in the aftermath. What will Britain, Germany and Italy gain?
Germany: Lorraine, Luxemburg, Morocco, Middle Congo and Gabon. Maybe some territorial adjustments between Germany and Belgium (eastern Wallonia to Germany, French Flanders to Belgium).
Italy: Nice, Savoy, Corsica, Riviera, eastern Algeria, Tunisia, Dijbouti.
Will the Anglo-German alliance of convenience hold or will it break up?
Both may happen, but it is more likely that it can hold. This war shall most likely make British strategic pendulum towards deeming the Triple Alliance reliable continental partners. To break the alliance it would require something truly stupid.
I don't think Britain will find good allies in Russia or France against Germany so maybe Splendid Isolation?
Unlikely, esp. because France is very likely going to slip into proto-fascist revanchism and Russia shall be most likely hostile or fall into revolution. And if Britain lost Canada to America while it was busy in Europe and Central Asia, it shall need strong allies.
France will be pissed off and prostrate at the feet of the Anglo-German combine and if this goes anything like WW1 or the RJW for Russia, they'll be facing serious internal upheavals at best, if not revolution.
Think of a crossbreed between Boulanger and Mussolini. That's what is going to happen in France.
Nicholas II will be forced to step down in that case in favour of his brother Michael if the empire even survives. I doubt either of them could be allies if they wanted to.
Indeed.
Where does the US stand in this if they remain neutral (or if they join and lose and get their Monroe doctrine stuffed up their butt)?
If they join, America is the only "Entente" power that is coming home with a decent result. Britain shall be forced to divert most of its potential fighting France and Russia, they shall not have a lot to spare to defend Canada. America has the time to prepare its forces (give them six months and they can raise an army that can swamp everything the distracted British Empire can deploy in Canada) and backstab northward. They have to fear Anglo-German-Italian naval reprisal after France and Russia have been vanquished, but its not going to happen for a while, they can entrench in Canada and offer a compromise peace to the war-weary Euroes (e.g. they give back Atlantic Canada and Quebec, and keep Western Canada, which is still very scarcely populated, and Ontario as well, if they go any good).
The Ottomans in the meanwhile will see their arch nemesis Russia crippled. Might they do better in the Balkans now that their Russian patron is preoccupied? They could perhaps hold onto Macedonia and Thrace IMO.
A reasonable assumption.