I only play on immortal, but I've had some near misses with cultural victory attempts, But never will I let one of those peace loving pansies steal the win from me. No I've never made a space ship(ok well i made 1 but only 1 in hundreds of games), and have absolutely no idea why a cultural victory even exists, but by that last screenshot You did almost the right thing.
A cultural victory is by far the easiest to shut down, razing only 1 city secures defeat(often capitulation) of the afformentioned civ. Your opportunity awaits when victory is just within the grasp of whichever waste of space is attempting culture domination. The Ai cultural strategy relies heavily on the slider which is great as the strongest unit the AI will have when nearing victory is infantry,if not rifles/sams. By this time if you have any kind of economy at all, you will be out teching them, and hopefully will have significantly higher production.
Keep checking the victory screen and as soon as you see a legendary, mount a naval based land invasion. I prefer modern armor/mobile artilery/with a few mobile sams. You can negate air support for the most part, as the mobiles are more than enough for this kind of enemy. In fact you really only NEED around 30 units in most cases to twart the victory, it wont win the war but it will stop you from losing the game Better yet bring 70 and raze all the cultural cities, or what I would do is bring 400 and raze everything on the continent. but you still dont need a nuke.
Nukes should only be used against AI trying for a space victory, and not in moderation either. Fight fire with fire, If they want to build a rocket to alpha centauri, build 100 rockets to them.