OK, finally finished one game at 1070 AD. A lot of mistakes made. Playing Peter I had 2 settlers from huts and a tech (priesthood), pretty good. There are a lot of floodplains, therefore all 3 cities would be able to hire 7 specialists (adding one wonder, that's 91 GA points/turn). Enough health resources and my total health was at 18, so no health problem. First GA bombed a city to get marble hooked, and copper is within another city's radius. Just no stone.
Each city had the same building sequence at the beginning: one worker, then granary whipped immediately at size 2 as planned. Then library, then theatre in every city as soon as available, then another worker.
After that, the capital would build Pathenon, another city with better hammer production would build sistine chapel, and the 3rd one would build national epic. Temples and monasteries, sometimes missionaries would be built while waiting for available techs. The capital would build all cathedrals.
I first tried to get philosophy ASAP and immediately went for GA runs. By around 300AD I had 15 GAs (plus one from Music), but that's only enough for 2 cities. The third city had less than 10k culture at that time and research was so slow that I only got liberalism in 425 AD.
Came back and restarted from an earlier save, and built 6 cathedrals in the capital while the other two cities running GAs. Wasted one GA on a city which already achieved legendary (the Ctrl key, ah!) Victory at 1070, so it could be 4-5 turns earlier if I did not make that mistake.
Neither of the above two strategies are optimal. Maybe the best way is to only build 2-3 cathedrals in the capital, then all 3 cities do the GA run. With 7 artists hired, it's easy to get 15 GAs in a relatively short period of time. Suppose the GA run takes 40 turns, and while running it, the capital has around 450 culture and the other two have 200 each, then at the end of it, the capital would have 18k culture, and the other two had 8k. Put 2 GAs in the capital and 7 each in the other two, and you are almost there.
If the above strategy holds, then it is not necessary to get all religions. You probably only need to found 4.