Catalyzes phosphorolysis of the pyrimidine nucleosides uridine, thymidine and 2'-deoxyuridine with the formation of the corresponding pyrimidine base and ribose-1-phosphate. phosphate + uridine = alpha-D-ribose 1-phosphate + uracil phosphate + thymidine = 2-deoxy-alpha-D-ribose 1-phosphate + thymine 2'-deoxyuridine + phosphate = 2-deoxy-alpha-D-ribose 1-phosphate + uracil Binds 1 K(+) ion per subunit. Homodimer. Belongs to the thymidine/pyrimidine-nucleoside phosphorylase family. Extended N-terminus.