Step 1: Think of UDP-glucose as a delivery van that carries activated glucose to several destinations. Identify which listed pathway it never visits.
Step 2: It delivers glucose for glycogen building, supplies the uronic acid cycle to form UDP-glucuronate, and participates in interconverting galactose. It is also used to attach sugars to proteins and lipids.
Step 3: The hexose monophosphate shunt branches off straight from glucose-6-phosphate and runs through 6-phosphogluconate to produce NADPH and pentose sugars. At no step does it require UDP-glucose.
Step 4: So the pathway that does not use UDP-glucose is the HMP shunt.
\[\boxed{\text{HMP shunt}}\]