Place gammon in a large pot; cover with cold water and bring to a boil. Add split peas and pork belly; simmer for 10 minutes. Skim off any foam that rises to the surface of the water.
Transfer gammon and pork belly to a plate using a slotted spoon. Strain split peas through a sieve, discarding water.
Place gammon, split peas, and pork belly back into the pot; cover with fresh water. Bring to a boil; reduce heat and simmer gently until gammon and pork belly are tender, 2 to 3 hours.
Transfer gammon and pork belly to a plate using a slotted spoon; cool until easily handled, about 10 minutes. Remove gammon from the bone and slice. Slice the meatiest parts off the pork belly.
Stir the chopped meat back into the pot. Add celeriac, leeks, onions, and carrot; simmer until softened, about 15 minutes. Add whole sausage link; simmer until heated through, 10 to 15 minutes. Remove sausage link and slice into small pieces; stir back into the soup.
Sprinkle celery leaves, salt, and pepper over soup. Simmer until celery leaves are warmed through, 1 to 2 minutes.