S-adenosyl-L-methionine-dependent methyltransferase that catalyzes the formation of 5-methyl-uridine in tRNAs and some mRNAs (PubMed:31361898, PubMed:33799331, PubMed:34556860). Mainly catalyzes the methylation of uridine at position 54 (m5U54) in cytosolic tRNAs (PubMed:31361898, PubMed:33799331). Also able to mediate the formation of 5-methyl-uridine in some mRNAs (PubMed:34123281).