Impact of Alan Turing's approach to morphogenesis
This is a very interesting question. Many people have researched this topic, and many still are. But regardless, I had never heard of Alan Turing's contributions, so thank you!

First of all, I cannot actually find who first coined the term morphogen. Though people had hypothesized that chemicals could play a critical role in development through much of the 20th century, I cannot actually find the first person to use morphogen. But the most important paper really came from a guy named Lewis Wolpert, who came up with the model of a gradient of morphogens leading to differential cell fates. The idea being that if some area of an embryo produces a morphogen at a very high concentration, then as you move away from that area, the concentration goes down. So if this morphogen is required at or above a certain threshold for activity, then only those cells with that concentration will have a certain cell fate, while at lower concentrations, the cells can become something different.

But this does not really answer your question. You are asking how a single cell, which is spherically symmetrical, can determine a particular axis. Though most organisms do this is in slightly different ways, the most common feature is that sperm entry point breaks the symmetry. The best way to explain this is to show you a diagram of Xenopus (frog) eggs.



Image from: http://studentreader.com/nieuwkoop-center/

The Xenopus egg, first of all, is inherently not spherically symmetrical. There is a black animal pole, and a white vegetal pole. The sperm can only enter a marrow region of the egg about 30˚ north of the animal/vegetal line. Upon fertilization, an event occurs where the pigmented areas turn toward the sperm entry point, leaving a gray crescent. Nearby the gray crescent, in the vegetal pole, a structure called the organiser develops. This organiser creates many of the morphogens that then pattern the rest of the embryo. 

Researchers have studied this a lot in many different organisms, but a few things really remain constant: eggs are not exactly spherically symmetrical, and the sperm entry point provides asymmetry.