Catalyzes formation of 5-hydroxymethyldeoxyuridylate (5HMdUMP) as a step in the pathway that replaces dTMP by thymidine hypermodifications in the viral genome (PubMed:34522950). As a final result of the pathway of hypermodification, 5-aminoethyl-2'-deoxyuridine (5-NedU) substitutes for about 30% of thymidines in the viral DNA (PubMed:34522950, PubMed:29555775). These modifications probably prevent degradation of viral genome by the host restriction-modification antiviral defense system (PubMed:34522950). (6R)-5,10-methylene-5,6,7,8-tetrahydrofolate + dUMP + H2O = (6S)-5,6,7,8-tetrahydrofolate + 5-hydroxymethyl-dUMP Belongs to the thymidylate synthase family.