Preheat the slow cooker if necessary; see the manufacturer's instructions. Heat the oil in a frying pan, add the sausages and brown on one side, turn and add the onion. Fry, turning the sausages and stirring the onions until the sausages are browned but not cooked.
Add the potatoes, tomatoes and black pudding to the slow cooker pot. Lift the sausages and onion from the pan with a slotted spoon and transfer to the slow cooker pot. Pour off the excess fat, then add the stock, Worcestershire sauce and mustard. Tear the leaves from the thyme stems and add to pan with some salt and pepper.
Bring to the boil and pour over the sausages. Press the potatoes down so that the liquid covers them. Cover with the lid and cook on low for 9–10 hours or overnight. Stir before serving and garnish with extra thyme leaves. Serve with slices of white bread or a poached egg.