In 1882 the French suddenly pulled out of the Franco-British planned intervention in Egypt. Gladstone government extended an invitation to Italy for a participation in the intervention (there were significant Italian economic interests in Egypt - although not on the scale of the French and British ones and quite a number of Italians living in Alexandria).
The Italian government declined this invitation, mostly for internal political reasons but also because Italian participation in the Austro-German pact was under discussion at the same time).
If the British invitation is handled better and the Italian government finds more problems with Austria it might be feasible for Italy to participate in the intervention and possibly become a (junior) partner in the Egyptian protectorate.
The most likely outcome would be closer ties to UK and no Italian participation in the Central Powers Pact. Stronger Italian intervention during the Mahdi insurrection would be also likely, and Sudan might become another Italo-British co-dominium. France might be unhappy about Italian presence in Egypt but I don't believe it would be a serious problem.