I can give some information on Great Britain and Russia. When he assumed power, Napoleon III did his best to assure the rest of Europe that his rule would be nothing like his famous uncles. He was quoted saying famously "L'Empire, c'est la paix." (The Empire Means Peace)

Some powers (such as Britain) were understandably skeptical while others (such as Russia) couldn't care less.

Let's start with Russia.

Tsar Nicholas, who saw himself as a guardian against revolution and a keeper of the peace of sorts in Europe, actually approved of Napoleon's coup d'etat and his treatment of the 'reds' during the revolution and his efforts in restoring order in France. (He himself lent troops to neighbouring Austria to crush a rebellion in Hungary in 1849). As the absolute ruler of Russia himself, he wasn't opposed to the military dictatorship that Napoleon III set up and he acknowledged him as a ruler of France. However Nicholas also had a firm belief that only God could ordain an imperial dynasty and he refused to accept Napoleon's Imperial claim as equal to his own. When Napoleon III became emperor, Nicholas was the only one in Europe who refused to address him as an equal - "monsieur, mon frere."

Nicholas was also known to have said regarding Napoleon III. "An Emperor without Divine Right is not an emperor, he will be an emperor in fact, but never an emperor by right." This seems like a trivial thing but it was considered an enormous indignation and insult to Napoleon, which would come into play later when Napoleon aligned himself with Britain.

Great Britain's initial reaction is what you'd expect. They had natural reservations about a Bonaparte on the French throne and many were outraged with how he overthrew (illegally) the French constitution much like his famous uncle. The famous massacre on the boulevards and the repression that followed with his government soon after didn't help.

Queen Victoria herself in particular wasn't a fan of Napoleon III the person. She and her husband Albert considered Napoleon III a "decidedly unsavory character" - a smoker, a gambler, a drinker, a leecher, with two bastard children and a string of mistresses. In fact their dislike of Napoleon III was one of the reasons she held out so long before giving him a royal invitation. The newspapers in Britain too (such as the Times) were equally hostile initially.

What changed that was the emergence of another belligerent power that was threatening Britain's interests in Russia. It also helped that Napoleon III made friendship with Britain a cornerstone of his foreign policy.

Long story short, the Ottoman Empire (at the encouragement of France) pissed off Russia when they granted equal status between Catholic and Orthodox clergy in the holy shrines in the Holy Land. Russia threatened war on the Ottomans and they in response, asked France and Britain to help. Britain (with an eye on Russian interests threatening their possessions in India) and France (who saw this as an opportunity to assert a renewal of French power and influence without a full scale war) became closer as a result of this conflict.

The common cause against Russia helped Napoleon's stock considerably with Britain. The Times started printing favorable articles on him, posters of Napoleon and his wife were displayed all over London. And as soon as France committed troops to fight alongside Britain in the Crimea, even Queen Victoria had to finally cave and consent to a royal invitation to Napoleon III. Leading to a further warming of diplomatic relations.

Source : Napoleon III and his Carnival Empire by John Bierman