Catalyzes the formation of 5-methyl-uridine at position 1939 (m5U1939) in 23S rRNA. S-adenosyl-L-methionine + uridine(1939) in 23S rRNA = 5-methyluridine(1939) in 23S rRNA + H(+) + S-adenosyl-L-homocysteine Contains three structural domains: an N-terminal TRAM domain, a central domain containing an iron-sulfur cluster, and a C-terminal domain that displays the typical SAM-dependent methyltransferase fold. The function of the iron-sulfur cluster remains unclear. It may be involved in the correct folding of the protein or may have a role in RNA binding. Belongs to the class I-like SAM-binding methyltransferase superfamily. RNA M5U methyltransferase family. RlmD subfamily.