Whew. Luckily I don't have the deluxe version and it loaded up fine. Here are my results.
My start wasn't that great, but I got extremely lucky and had all 8 cultural CS in the game. When I realized that, I started taking some notes.
By turn 135, I had all 8 CS allied, and I had Biology so I was generating 230 culture at that point (160 from the CS, 70 from my cities).
Turn 150, I built the Hermitage and was generating +280.
Turn 180, I finished the Louvre, and by turn 189, I had all the necessary techs to finish (Radio + Telegraph). At turn 193 I bought my broadcast tower and was generating 459 Culture, and I finished Cristo at turn 195 (with the help of a Great Engineer, I believe). Still, it took me until turn 206 to get all my culture complete, and because of lackluster production in my cap, it took me 13 turns to finish the Utopia Project.
When I reloaded my game tonight, I was thinking of ways to increase the Utopia Project speed. 1) I realized I could have captured and annex Novograd and buy a work boat to well one of my offshore oils. 2) It wasn't until several turns later that I realized I could've grown some more population because I never built a Granary and I wasn't allied with 1 maritime state. I also noticed that there's a bug with my save, as if I remove all my engineers, I get -1 hammers due to specialists. However, it also would have been kind of dirty if I had done better because the patch added +1 hammers from engineers. Either way, I think if I was at the top of my game, I *might* have been able to shave one turn off, but still, that would've only been turn 218.
I don't think I have time to do another game, but I was thinking -- if I were to use the exploit of selling cities to other civs, how fast could you win with Liberty+Rex-> Save up culture, sell your cities and win.