(I’m going by memory here, so sorry about any mistakes)
In the short term, it is reasonable to suggest that – assuming a near-complete victory over the British and Prussians – Napoleon would be defeated by the Russians and Austrians. The French were in no state for a prolonged war and the victory at Waterloo would have to be very one-sided for them to face their next set of enemies without being significantly weaker and the enemies being (probably) stronger. It’s unlikely Napoleon can just keep on winning battles – realistically, he still loses power in 1815/6 regardless.
However, if he is defeated in late 1815 by the Russians and Austrians, the post-war scene will be very different. It will be the Russians who will hold the strong hand, rather than the British. It will be the Russians who will take Napoleon into custody (assuming he doesn’t flee or gets killed in battle) and it is they who will set the early course of events. The temptation to meddle, to appoint Louis’s cousin as king of France (and effective puppet) or even to do the same to Louis himself will be irresistible. The occupation will be a great deal worse in this timeline, as the Russians avenge themselves on the French and generally establish themselves as top dogs in Europe. The British will not be happy, obviously, but it will take time to replace the losses at Waterloo or for economic pressure to take its toll on Russia. This will give the Russians a short-term chance to order Europe to suit themselves, with the various kingdoms and suchlike brought into a loose alliance. Depending on how they play it, this could work fairly well in the short term.
(Ironic off-shoot – Napoleon flees to Britain, which eventually finds itself using him as a tool to undermine Russian-dominated France).
It will probably also start to fall apart fairly quickly. The French will not take kindly to being kept under a Russian-supported government. There were a lot of tensions between 1814 and 1815 that will reappear, if steps aren’t taken. Prussia, too, will not be happy taking orders from the Russians. Given time, the alliance will start to crack at the seams – perhaps, in this timeline, there will be a resurgence of antimonarchical feeling right across Europe, backed by Britain. I think there would be another major war, as soon as the horrors of the last one are forgotten. Would this count as a ‘first’ world war?
Chris