What does the applicable code say?

If the duct is exposed in the space served, you usually don't have to. There can be spaces with high humidity, though. so it depends. But usually everything not in the space served has to be insulated.

The code doesn't say much just what insulation to use it's the specs here that say what the duct needs but it's a vague description. In saying that the temperature will be constant in the supply from the AHU at 12.8C when it hits the VAV with a reheat coil. My understanding is the supply from the AHU is ambient and doesn't need insulation and the air after the VAV box get's insulation.

When the air leaves AHU, it will be cooled or heated to a constant temp (12.8°C or 55°F as you said). The airflow of the VAV boxes should be designed based on 55°F - 57°F inlet temperature.

Assume what will happen without duct insulation: supply air temperature might be heated to 65°F in summer at the VAV inlet. When at peak cooling conditions, you may not achieve the target room temperatures.

In general, all supply and return ductwork in unconditioned spaces shall be insulated.

So I can assume that as soon as the air leaves the AHU supply through the building to reach the VAVs it is running through unconditioned spaces including the mechanical room and needs insulation.

The code WILL tell you, the energy code for your area. The energy code I use doesn't technically require insulation inside the thermal envelope of the building except to prevent condensation. Find your energy code.

Your VAV's are cooling the air? That doesn't sound right to me. Are you sure the cooling isn't taking place at the air handler?