Whilst I'm tempted to just link to my timeline and leave it at that...
The problem is that the National Guard [essentially a localised militia] and the French Army [a professional and conscripted military force] were very different. The Prussian siege of 1870 had left Paris cut off from the rest of France and whilst this bred the circumstances of the Commune it also meant that the ideology did not really spread very far.
Thiers and the Republican Government were quite canny with how they approached the Commune - they brought soldiers from the provinces who were more skeptical about Paris and the sort of radical socialism that the city represented. Men who were less likely, they thought, to be tempted to switch sides.
One of my PODs was to remove Thiers, have the monarchists triumph in the election, and thus make the republican 'soft' left more ambivalent to the Commune through their shared anti-monarchism and anti-catholicism. Really, though, you need the Commune's leaders to be more effective and more united in the early days of the revolution. They need to hold the outer forts of Paris, they need to be more organized in resisting the anti-Commune strike that was coming, and they need to be better at forging links with the other insurrections bubbling in places like Lyon, Marseilles, and St Etienne. If they can win the Battle for Paris then who knows what might happen next.
As for what post-1871 France would look like, I'll just say read my timeline.