Catalyzes formation of 5-hydroxymethyldeoxyuridylate (5HMdUMP) as a step in the pathway that replaces dTMP by thymidine hypermodifications in the viral genome (Probable). As a final result of the pathway of hypermodification, 5-Nalpha-putrescinylthymidine (Nalpha-PutT) substitutes for about 50% of thymidines in the viral DNA (Probable). These modifications probably prevent degradation of viral genome by the host restriction-modification antiviral defense system (Probable). (6R)-5,10-methylene-5,6,7,8-tetrahydrofolate + dUMP + H2O = (6S)-5,6,7,8-tetrahydrofolate + 5-hydroxymethyl-dUMP Hydroxymethylase appears immediately after infection. Belongs to the thymidylate synthase family.