The Shadow of the Fashoda is a good thread on this subject .....
Along with the previous posts, the French would be screwed in the long run. Perhaps very screwed, losing an empire, having it carved up among the world powers.
As you state, the French might be able to capture the Sudan in the short term. But long term, the British navy would basically have the French forces dying on a vine as they would most likely be cut off.
And then perhaps a feeding frenzy would start on the French empire.
the Japanese might ally with the British and take over French Indonesia if the British let them.
The Germans/Spanish/Belgiums could ally with the Brits and carve out more of Africa from the French and take whatever the Brits would allow them to have.
Would the Italians join in the carving of the French turkey to try to take Corsica and Tunisia? Tunisia probably yes, Corsica probably not since this is a grand colonial war and Corsica is too close to the French homeland. I do not see the French, British or anyone else wanting to escalate this from a colonoal war.
What would the Ottomans, Austrians, Russians do? Would they stay out of it, go at each other. The Ottomans might side with Britain to gain Djibuti and/or Somalia (once again whatever the British will let them have). The Austrians would probably stay out, The Russians might use the war for another go at the Ottomans or use it to cover activity in the Far East.
The US had an imperialistic tone at this time as well. Teddy Roosevelt would probably keep the USA out of the European argument. But he could use any effect on neutral shipping as a precursor to join one side or the other. Say the French accidently or not, sink an American supply ship to Britain or one of Britians colonies. US uses this to ally with Britian and carve out the French West Indies and any other Pacific island that they can get their hands on. Once again it is what the British will allow them to take. Or would the US be stupid and take on the Brits to take Canada? (I believe that the Yanks were over their lust for more northern territory at this time and Canada was independent as well, but you never know). Would the US and Venezuela team up and move in on the British in Guyana? (once again stupid, but who knows) Perhaps the US can gain territory without firing a shot by simply buying islands from France.
This would be a colonial war. After the French loose their Empire, who knows what would happen? Would they turn socialist or ultra-nationalist? There would probably still be another war in Eastern Europe in a decade as the Archduke of Austria could still be assassinated. But it could be Russia and Serbia against Germany, Austria, and the Ottomans joining in to fight the Russians. The French might rebound, build up their military and have a go at the Germans, but perhaps not. The Brits most likely would stay out letting the Austrians have Serbia and then watching them implode and pick up the pieces of the smaller countries through allainces. (This is covered better in te Shadow of the Fashoda thread)