Medium answer: Kinda. The chemical gradients cause a signaling cascade that modifies transcription of DNA (it's really complicated). This transcription change then causes the cell to become a XXX_neuron. However, there are many many waves of this process occurring with a lot of cell death along the way. When those cells are not the 'final' cell of the nervous system, these transcriptions can cause further and more complicated chemical gradients to exist in the fetus. These complicating recursive loops can also self-affect cells and cause them to change yet again.

We're still discovering a lot here.

Also, this is largely how ALL cells in a body work, not just neurons. Be careful though, this is very very complicated stuff and everything I've written has a caveat.

Kinda like asking if gravity and fluid dynamics are encoded in the blueprint for an aircraft.

The design relies on them, and exists in the form it does because of them.