Catalyzes the phosphorylation of nicotinamide riboside (NR) and nicotinic acid riboside (NaR) to form nicotinamide mononucleotide (NMN) and nicotinic acid mononucleotide (NaMN). The enzyme also phosphorylates the antitumor drugs tiazofurin and 3-deazaguanosine. ATP + beta-nicotinamide D-riboside = ADP + beta-nicotinamide D-ribonucleotide + H(+) ATP + beta-D-ribosylnicotinate = ADP + H(+) + nicotinate beta-D-ribonucleotide Optimum pH is 6.5-9. Cofactor biosynthesis; NAD(+) biosynthesis. Monomer. Belongs to the uridine kinase family. NRK subfamily.