In a 5-to-6-quart slow cooker, place celery, carrots, onion, potatoes, and thyme.
Place corned beef, fat side up, on top of vegetables and sprinkle with pickling spice; add enough water to almost cover meat (4 to 6 cups). Cover and cook on high until corned beef is tender, roughly 4 1/4 hours (or 8 1/2 hours on low).
Arrange cabbage over corned beef, cover, and continue cooking until cabbage is tender, about 45 minutes (or 1 1/2 hours on low).
Thinly slice corned beef against the grain and serve with vegetables, cooking liquid, and grainy mustard.