Catalyzes the formation of UDP-glucuronic acid which is required for capsular hyaluronic acid synthesis. H2O + 2 NAD(+) + UDP-alpha-D-glucose = 3 H(+) + 2 NADH + UDP-alpha-D-glucuronate Nucleotide-sugar biosynthesis; UDP-alpha-D-glucuronate biosynthesis; UDP-alpha-D-glucuronate from UDP-alpha-D-glucose: step 1/1. Belongs to the UDP-glucose/GDP-mannose dehydrogenase family.