The Brits are likely going to have to start the Somme offensive a lot earlier and will give it a lot more priority. The French army will need the rescue and the circumstance will be a lot more dire - I can definitely see it having to start a lot faster than it did.
The buildup is definitely gonna be telegraphed. So will Falkenheyn determine it's a big enough threat he has to cut off possible reinforcements for Verdun to hold the line? If the French are throwing everything they have into the gap they can at least slow it. The idea is to bleed the French, nto take territory, so if they counterattack I can see the Germans just digging in and holding - or as the reserve divisions have to be sent to the Somme instead.
Also a possible change for the German doctrine. At this point their doctrine was to as soon as a trench was breached counterattack. If it was a probe, if it was an assault, if it was a raid.. Counterattack. Drive them out immediately or slow them down (the theory was). Instead it gave up a lot of the benefits of the trench and opened them to nasty casualties.
Them doing better here also means that may not be standard doctrine, so they might not do as /well/ on the defense but they won't be bled as bad either.
Also dunno if Foch will decide to evacuate Nice and pull back to a better defensive line. He seems to be prioritizing Verduna nd hasn't put up a 'not one more centimeter' call so there doesn't seem a political imperative to hold both no matter what. So if he pulls back and mostly levels the city, the Italians get little from it beyond more land - a big morale boost, but little else. Fall back to better defensive terrain, dig in, the Italians have to build railroads, bring forward supply dumps, etc. They won't be heading forwards again for months. Transfer the divisions northward.
French yelling for help also virtually guarantees more of the RN being sent tot he Med. They can easily spare a Battleship division or so - another 6-8 BB's and cruisers won't strain the Home Fleet too much and they'll still have a minimum 2:1 supremacy over the Kaiserlichemarine. That lets them lock down the Med and still threaten Italy's coast with bombardment. The RN is still much, much better than the French and thier ships will outperform the Italians/AH. So that just puts it again as a stalemate where both sides patrol aggressively. The AH won't deploy thier own fleet so far forwards against the RN, that makes the Med a relatively safe zone (beyond all the submarines). The Germans could always go for something exotic like sending some of thier submarines over the rail line to be reassembled there that they'd thought of a few times over WW1 but never did?
In the Balkans Bulgaria is gonna be demobilized, Greece is neutral (but ould have a pro-CP government). But Greece being neutral also means no Salonika, and no weird dispatch of divisions to there (was this already commented on?). Romania (even if neutral) will sell lots of grain to the CP. WIth Russia being the only active front likely this will solve all the food issues. Could even have the Germans sending some things down to the Ottomans to help (getting them to the front is still an issue with the nonexistance of the Ottoman rail network). Ottomans can push for Baku or something?
The Kaiser will want the navy to get in on the glory as well and might even let them get aggressive. This will likely not go well given the British had pretty well broken thier ciphers. But hey, north sea naval battle! If the French are being desperate, and the British are being slaughtered in the Somme the Kaiser might tell them to go out and sail for victory to push them over the edge.