To get the Olympics revived, you need a relatively short communications and travel time between participating nations. I think that you could pull it off technologically for Europe in the 1850's, when the telegraph was in widespread use.
Getting it going worldwide will require practical trans-Atlantic communications--meaning more than one or two working cables. Fast steamships will also be key here.
My logic here is simple--not sure if it's valid:
Athletes can't be taking months to get there and back again, especially amateurs
A short communications loop is needed to keep up interest, as well as to create the idea of nations as neighbors.
The Greeks managed it without 1850s technology!
Seriously though, even in 1896 "global" participation was barely above token levels. If it starts earlier the emphasis probably wouldn't be on its global nature, but on more generalized prestige. Initially, it would have to be a mostly European affair, and probably more specifically regional than that.
If Louis XV makes it a courtly event, it's going to be mostly French allies who show up. If the Hapsburgs do it on ethnic lines as a way of keeping their subjects content with their rule, there might be some interest from their neighbors but probably not much beyond that.
And while it could easily spread beyond that to a different court every time the games are held, they're not going to leave Europe for centuries (especially if Istanbul is considered Europe, which is the only- and still pretty unlikely- possible exception). They might not turn away the odd ambassador or adventurous prince who likes competition, but it's only going to be global in the timeframe NHBL mentions.