Well it was a long slog but I eventually reached 20,000 culture for victory in AD 1974, scoring Firaxis 2764, Jason 2945. Quite a small score given 67 hours played, but it was my first cultural attempt, and hopefully I'll get my first award- for lowest scoring 20k.
By the end only America, England and Spain were left, but they were still some way from space or 100k. My final army consisted of 173 modern armor, in case a rival threatened victory. Interestingly I stayed in republic throughout the game- the unit support in democracy put me off switching.