Catalyzes the phosphorylation of nicotinamide riboside (NR) and nicotinic acid riboside (NaR) to form nicotinamide mononucleotide (NMN) and nicotinic acid mononucleotide (NaMN). ATP + beta-nicotinamide D-riboside = ADP + beta-nicotinamide D-ribonucleotide + H(+) ATP + beta-D-ribosylnicotinate = ADP + H(+) + nicotinate beta-D-ribonucleotide Cofactor biosynthesis; NAD(+) biosynthesis. Monomer. Belongs to the uridine kinase family. NRK subfamily.