Peter Ackroyd writes on page 88 of his account of the English Civil War:
Cobblers were well known for their radical Protestant sympathies.
What was so special about cobblers?
Cobblers were workmen whose customers wore shoes, and therefore traveled more than most others (only a few people wore shoes in the Middle Ages). By definition, these were the few people that "got around" and knew enough about conditions in other parts of the country to start a revolution, and cobblers, as providers of "personal services," were their confidants. This was at a time when there was no TV or radio, there were barely "newspapers" (in a few world capitals), and most of what passed for "news" was spread, word of mouth, by travelers. A cobbler would have access to many of them.
Moreover, people with shoes were the ones with the better chances to flee successfully if a rebellion went bad, and cobblers tended to be loyal to their (few) customers.
Here is an example of cobblers' roles in a French revolution.
The question occurred to me, why cobblers and not e.g. saddlemakers? The answer is that horsemen were mostly members of the landed gentry, a much higher social class than most "shoe wearers," and saddlemakers were less likely to be their confidants than cobblers for upper working class people.
In the case of the English Civil War, the successful Puritans drew most of their support from independent "yeoman" farmers and urban artisans. These were the people (among the poor) that tended to wear shoes, and were "connected" to cobblers. (The landless farm workers and lumpenproletariat were mostly apathetic, but to the extent that they had a political preference, they tended to support "King and Country," that is Charles I.) It's usually not the hopelessly downtrodden, but those one or two steps up the economic ladder (and are afraid of being pushed back) that tend to be revolutionaries. That status pretty much describes "cobblers." Such people were also followers of the Protestant ethic, which in this case set them against their Catholic King.