It will be difficult to get even time separation for the LED turn-ons if you try to use zeners. The exponential nature of the cap charge, combined with the ever increasing load as the zeners conduct, exacerbates this problem. if you want to pursue an all-discrete solution, you can use a current source to charge the cap, buffer it with an emitter follower, and use cascaded emitter followers (or diodes) for the level shifting. Emitter followers seemed to have slightly better linearity in the simulation. I have posted a schematic below.
I think that an LM3914 would be simpler, if the all-analog solution is to be pursued. I have posted a possible solution below, which has not been tested or simulated.
A 555 and a 5- or 6-bit shift register, or a 555 with a binary counter and some decoding logic, is also a possibility. A PIC would certainly be simpler from a hardware point of view, except that it requires a voltage regulator. Any digital solution would have to deal with the noisy power supply of a vehicle. The analog solutions have an advantage in this respect.