Interesting - a "limited" Great War
As per the title. I understand that the German military establishment had little confidence in the Schlieffen plan, and that there was a great deal of uncertainty over whether Britain would actually intervene in response to a Belgian invasion. So for the sake of this scenario, let's say that Britain makes it's objections clearer. Even then their will be an anti-German constituency in Britain... but OTOH the cabinet was narrowly divided on whether to enter even with the Belgian invasion. So in addition to not invading Belgium, let's assume the Germans make an active and successful effort at avoiding a cassus belli that would put wind in the sails of the anti-German faction.
Of course the Germans without Belgium to pass through would be unable to launch a meaningful assault on France proper. But OTOH France would be similarly unable to assault Germany, le.aving Germany free to tear into Russia. And, importantly, a long drawn out war with France would be far less problematic for the Germans in this scenario. On the A-L front they could simply hold a (mostly)defensive line and let the French throw their troops into the meat grinder. The greatest source of urgency for the Germans came from the British blockading the German access to overseas trade(for example, nitrates). But the French couldn't perform a comparable blockade given their inferior navy, so the Germans could safely focus on Russia and let the French beat themselves bloody on the trenches until they gave up.
However, a static front in Europe doesn't mean the French and Germans couldn't duke it out at sea or in the colonies. And so we've arrived at the title
question: how is this likely to proceed?
Looking at a 1914 map of the world, the only direct borders between the German and French empires are between Cameroon-French Equatorial Africa and Togo-French West Africa. The latter looks completely indefensible at first, being swamped by FWA... but on further consideration I can't imagine the supply chain through FWA would be developed enough to be of much initial use, which means the French strip between Togo and Nigeria is in much the same troubled position of having to be supplied by sea via a narrow stretch of coast.
The French would probably have a first mover advantage though, due to having a larger number of troops in West Africa and far more colonial resources to draw upon. So it becomes a question of whether the Germans are able to rush forces to Togo and or Cameroon in time to shore them up(and whether they'd bother). Beyond that I'd imagine any assaults on other colonies would target strategic islands and ports... although if the French are frustrated by their inability to confront Germany in Europe, they might shift full bore into invasions of German colonies, with Germans feeling compelled to match them. The downside for the French here is that the German navy clearly outclassed their's, so if they were smart they'd limit direct engagement as much as possible.
One thing I've overlooked. Britain not intervening doesn't mean nobody else will. Presumably the Ottomans still side the Germans against their Russian nemesis... and I could see the French launching a massive assault into the Levant in lieu of having a worthwhile front in Europe... which might actually mean the Ottomans being in a worse position then they were OTL facing only what France and Britain bothered to scrape out of the barrel. Then there's Japan and Italy, which both face an interesting choice.
For Italy, their craving for the ethnically Italian A-H territories still stands. But at the same time they had designs on Corsica, Savoy and French North Africa. OTL they waited for nearly a year whether out of uncertainty or just to haggle for a better deal, but realistically they would never have backed the CP when that meant facing off against the British navy. Since that's not a concern in this scenario, they may give more thought to the possibility of joining the CP.
As for Japan, they were principally Britain's ally and Russia's nemesis. Without the British participation in the war, I imagine joining the CP looks rather tempting- on the one hand they might snap up the German concessions in China and some German Pacific islands. On the other hand they might snap up the French concessions in China, French Pacific islands and Indochina, while weakening Russia and grabbing Russian Sakhalin and Manchuria in the bargain. The latter is higher reward, but also higher risk- the French in Indochina and Russians in Siberia will be far tougher to overcome then Germany's limited Pacific possessions.
Of course, Britain not entering the war doesn't mean they're neutral. I imagine the French would have access to the Suez canal while the CP would be denied, with the excuse being that the French are half-owners of it. And they might well lobby the Japanese and Italians to "not pick the wrong side".
Interesting - a "limited" Great War...
The French, after the initial Plan XVII battles, showed themselves quite capable of adjusting to a more defensive strategy, and if the British are essentially friendly neutrals, seems likely they (the French, that is) would avoid simply grinding away; one possibility - as opposed to fighting some sort of "umpteenth battle of the Isonzo" effort back and forth in the Moselle-Meurthe-Saar region would be to make an effort east of the Vosges, driving northeast from Belfort toward Mulhouse and the upper Rhine.
It would not be easy, but the French mountain troops certainly seem to have been as capable as their German and Austrian enemies, and it makes for less of a campaign of artillery and more of manever - at least in the high country.
The other possibility is, as you suggest, are a maritime war, which is an interesting point of departure. The French had overwhelming superiority at sea in 1870-71 and even attempted to blockade the North Sea German coast, but had little success, largely because of the requirements of coal burning navies for replenishment in harbor.
That's far less likely because of the much stronger German navy, but at the same time, the HSF can't really try to move into the Channel for fear of British reaction to German squadrons off the coasts of the Netherlands and Belgium.
The French can certainly close the Bay of Biscay to German traffic, and the Germans can close the eastern North Sea to the French, but that's about it. The French had four modern dreadnoughts in commission, with six more under construction, plus 12 large semi-dreadnoughts and six predreadnoughts, plus a few even older ships in reserve. Not exactly the Grand Fleet, but certainly enough to give the Germans pause about any operations outside of the North Sea, including trying to force the Channel, even with their battlecruisers...the Germans, of course, had a much stronger force of dreadnuughts, but since the entire point of the HSF was to deter the British, risking much of it against the French would be questionable in the event of even a neutral Britain.
The French could try a very distant blockade of the western North Sea, but the French cruiser forces are not especially strong.
The secondary element is, of course, a colonial-maritime war, and the French are capable of overwhelming the German colonial empire, given the general balance of access and the stronger French colonial forces. Togoland will go quickly; the Allies took it with a few thousand men from all sides and the French could muster the equivalent, using Dahomey; given the approach, the Germans can probably defend a little longer, but not for that long. Presume they surrender in August or September.
Kamerun will last a longer; as it was, it took the Allies 18,000 men and until the final German post surrendered in March, 1916. It will take time for the French alone to assemble an equivalently-sized force, so the Germans might last well into 1916 in Kamerun.
Southwest and Southeast Africa would both be daunting, even with the French position in Madagascar; probably simply a blockade and a guarantee from the British in the interim before a major expeditionary force could be assembled by the French there, sometime in 1915 or even 1916. At that point, probably a French campaign in Southeast Africa, but Southwest Africa is simply blockaded by the French.
The German Pacific territories are interesting; German Samoa could be overrun fairly quickly, and NE New Guinea and the Solomons eventually, in 1914-15 by expeditionary forces from French Indochina and Polynesia. The German territories in Micronesia and China would be more challenging, and a Japanese interest would be high. The US might negotiate for a purchase, simply to guarantee the sea lanes from Hawaii to the Philippines, but whether that could forestall a Japanese military operation is an open question.
The joker in the deck in the Pacific is the German Pacific Squadron, of course; as it was, von Spee et al dodged most of the Allied sea power and destroyed Cradock's RN squadron off Coronel, and the French would be very hard put to get the equivalent of the Allied forces historically needed in the theater out to the Pacific in a timely fashion.
The same question has to be asked of Souchon's squadron in the Med; one might expect they'd have been pulled out before the balloon went up, or they'd run for Austrian harbors, at which point the French would be pushing the Italians hard to enter the war on the Franco-Russian side. The Turks might never enter the war, actually; they have no formal alliance with either side and little to gain, especially if the British don't seize the Turkish dreadnoughts and the Germans are not in a position to influence the Turks.
The French would certainly make the navy more of a priority than it was, historically, and so that might also push them toward a defensive strategy; fewer heavy guns for the army, obviously. They may also sound out the British and US about purchasing existing ships; certainly that would provide a quicker method to bolster the Marine Nationale than waiting for the French yards to get the battleships under construction into commission.
As it was, "modern" French cruisers (defined as commissioned since 1894, which is sort of a push anyway) amounted to 20 armored cruisers of various classes (dating from 1894 all the way to
Edgar Quinet and
Waldeck-Rosseau, commissioned in 1911) and a dozen elderly protected cruisers, plus about five even older armored cruisers, in second-line service or reserve.
To be charitable, call it 36 ships, with the 12 most modern armored cruisers reserved for the French Atlantic (Channel) Fleet; that leaves 8-12 older armored cruisers and as many protected cruisers for West African, Southeast African, and Pacific waters, facing a German force of two modern armored cruisers and six modern light cruisers; the French may decide to send out some of their older predreadnoughts to back them up.
All things being equal, if the French are conservative on the "Western Front" one would expect the colonial war to drag out to 1916 or thereabouts, but the French would win the various campaigns. Distance and position is in their favor, given the German and Austrian preoccupation with Russia...
Best,