Again, now that we have this number, I think it's time to set our eyes on more plausible solutions -- such as not starting on a Plains Marble Hill with Grassland Deer in the Great Plains.
OK. Let us modify the assumptions and post the question again to everyone:
What is the minimum number of turns for 6 cities?
The question is obviously depending on your starting map, but let us assume you have an "optimal" start:
(0) You cannot use Plain Marble Hill; you cannot use Grassland Forest Deer. You can have deer; but not on Grassland Forest. You can have marble, but not on Plain Hill.
(1) You can have anything else you want; if you want to chop, there are enough forest tiles; if you want production, there are cows (2), mines (2), irons (1), copper(1), marble (1), stone (1); if you want to use slavery; there are two corns; ...
(2) Cities do not need warrior;
(3) Any size of cities is acceptable;
(4) Cities can have minimum distance;
0x000x0
0000000
0000000
x00x00x
0000000
0000000
0x000*0
(5) You obviously need workers; you can make as many workers as you want.
(6) Select any civs you wish (some civs give you + 25% for workers, some civs give you +50% for settlers);
(7) No advanced start.
(8) You start with 0 workers; you do have to build workers if you need them;
(9) Ancient era, starting with 1 settler and 1 other unit.
(10) Fast speed.
(11) Second city can produce settler also.
Can you list your answer and post strategy please?
Based on the discussion earlier, we know 1F4P start is important, which requires you to settle on plain hill and work on grassland hill forest(1F2P) or plane forest (1F2P), i.e 2F2P city + 1F2P.
Plane hill city (or parking on elephant) (2P) plus grassland hill forest (1F2P) ==> 1

4

.
Expansive boosts it to 1

5

.
You get the first worker in 40/6 = 7 turns.
Plane hill city (or parking on elephant) (2P) plus grassland hill forest (1F2P) ==> 1

4

.
Imp boosts it to 1

6

.
You get the first settler in 65/7 = 10 turns.
So you have 2-turn delay in both cases. This delay obviously significantly increases the power of workers so worker-first will likely to win.
Beside 3F4P-start, you actually can get 2F4P-start; but people will complain again it is not realistic enough.