To be clear, it is not impossible. Napoleon III and France might well support this project as part of a plan for a general reorganization of Europe along presumably pro-French lines.
At that time, funny as it may sound, Little Nappy was trying to maintain partnership with AII as a tool for achieving his goals in Europe while trying to placate the pro-Polish sentiments in France by making the anti-Russian
noises. Needless to say that he failed on both accounts as he had change to find out later. The British goals had been even less clear and, as far as I can tell, were mostly a pure demagoguery (to be fair, not up to the Palmerstonian insanity level).
On a pure practical side the “project” was fantastic by a number of reasons:
1. Little Nappy already had enough of experienced of the joined French-British operations during the CW when France suffered huge losses for pretty much nothing except for him gaining an appropriate addressing from AII.
2. While during the CW the allies had a numeric advantage at the theater (most of the Russian troops had been staying on Austrian border) and certain logistical advantage due to the underdeveloped roadmap of the Crimea and Southern Russia in general and the theater being far removed from the production centers of the Russian Empire, here the attempt of a meaningful landing would be just in the completely opposite area: concentration of the most of the Russian armed forces, closeness to the production centers and much better communications on the
Russian part of the theater (but not in a big part of a rebellious area).
3. The upraising at its top point (May 1863) amounted to 50,000 insurgents after which the numbers had been speedily decreasing (10,000 in September and after this single thousands) while the Russian numbers had been steadily growing (130,000 in August, 180,000 in October, and all the way to 220,000 in 1864). By the time the allies could realistically start implementation of the “plan”, they’d have a hard time looking for those whom they were intended to help, taking into an account that at no point they formed anything close to an unified army and even in May 1863 had been spread all over the internal areas of Poland and Lithuania. So, in the practical terms, the allies would have to fight a war on their own with a token Polish presence. Even from the very beginning the uprising was not a
popular one: the peasants were not interested and even promises of the polish “government” did not change this.
4. Even during the CW the allied achievements on the Baltic were close to zero and now they would have to capture a major port, hold it and greatly increase their own numbers because they’d have to fight a long war of exhaustion.
Getting France to commit wholeheartedly to this project, at a time when it has extensive overseas involvements, would take some doing, while getting Britain to join France in this project would take more.
That’s a very realistic assessment of the situation: making speeches and waving the fists is one thing but getting into the major war is a completely different issue.