Dye is a colour applied to hair or fabric.
Wool, as you no doubt know, is the hair of a sheep used to make clothing.
Are we getting any closer to meaning of this idiom?
Somebody who is dyed in the wool is uncompromising with fixed opinions that they will not change.
For example he’s a dyed in the wool conservative or a dyed in the wool Manchester United fan.
Are you dyed in the wool?
Want to dye some wool? Here's how!