Does gum disease spread to the palate if people don't look after their teeth?

Possibly. If you develop a periodontal abscess on the palatal aspect of an upper tooth, the swelling could involve the palate in that area.
Not usually. The cause of the problem is the bacteria on your teeth. So the gum inflammation usually is just around the areas of the teeth and doesn't spread into the roof of your mouth.
BUM DISEASE. It can but depends on what the cause of gum disease is, if not treated it can effect the bones under the teeth and in extreme cases mouth cavity can be involved due to ovehelming infection.There are various grades of gum disase and diffrent causes.
Infections spread. Gum disease is usually local to the surround bone and gums. However, if left untreated you can develop an acute periodontal infection that can spread not only to your palate but to any place in your body. Although relatively rare, it is possible. All infections have this potential and that is why prevention and early treatment is so important.
Periodontitis. Gum disease is a bacterial infection that invades the tooth's attachment to bone. As it progresses, more bone is lost as the infection spreads into the tissues. Palatial tissues will become inflamed adjacent to the teeth, but gum disease doesn't invade soft tissue such as yeast or an acute dental access.