Catalyzes the phosphorylation of thymidine monophosphate (dTMP) to thymidine diphosphate (dTDP), the immediate precursor for the DNA building block dTTP, with ATP as the preferred phosphoryl donor in the presence of Mg(2+). ATP + dTMP = ADP + dTDP Pyrimidine metabolism; dTTP biosynthesis. Homodimer. Belongs to the thymidylate kinase family.