Catalyzes the conversion of uridine into UMP and cytidine into CMP in the pyrimidine salvage pathway. ATP + uridine = ADP + H(+) + UMP ATP + cytidine = ADP + CMP + H(+) Pyrimidine metabolism; CTP biosynthesis via salvage pathway; CTP from cytidine: step 1/3. Pyrimidine metabolism; UMP biosynthesis via salvage pathway; UMP from uridine: step 1/1. Belongs to the uridine kinase family.