The idea of helping Sardinia in a second war against Austria was very unpopular in France to the point that Napoleon III did not have any kind of internal and external support other than his own whim, some collaborators and, of course, Sardinia.
Great Britain on the other hand saw badly altering the current balance of power of the continent (as always xd).
Well, the unpopularity of the War Napoleon started in Italy was directed to him; Sardinia had not irritated France in any way, the French people were annoyed at their ruler for sending their sons to die for a cause that had nothing to do with France (IIRC the whole Nice and Savoy thing was basically because Napoleon needed some gain to soothe discontent). So, in the scenario where Napoleon is not moved by Orsini's letter (which is possible), the French public has nothing to be irritated about with Sardinia. As for Britain, I will just quote from
here :
"The British government by 1859 was understandably eager to find some solution to "the Italian Problem." The peace of Europe, it was argued, depended upon the tranquility of Italy, and the independence of the Italian peninsula was necessary to the European balance of power. British policy, in practice, had
sought to isolate Austria on the Italian question in order to facilitate her complete expulsion from the peninsula. A widespread British belief was that Austria's Italian possessions were "a very questionable element of her real power" and that it would be "an unmixed good" if the Austrians "were driven out of Italy for good and all." The Palmerston-Russell cabinet, which was formed only two months after hostilities had erupted between Austria and the French and Piedmontese allies, quickly concluded that the only possible outcome of the war must be "the entire and absolute relinquishment by Austria of her Italian provinces."