1,2,3-triazoles are versatile building blocks with growing interest in medicinal chemistry. For this reason, organic chemistry focuses on the development of new synthetic pathways to obtain 1,2,3-triazole derivatives, especially with pyridine moieties. In this work, a novel series of 1,5-disubstituted-1,2,3-triazoles functionalized with pyrimidine nucleobases were prepared via 1,3-dipolar cycloaddition reaction in a regioselective manner for the first time. The N1-propargyl nucleobases, used as an alkyne intermediate, were obtained in high yields (87–92%) with a new two-step procedure that selectively led to the monoalkylated compounds. Then, FeCl3 was employed as an efficient Lewis acid catalyst for 1,3-dipolar cycloaddition between different aryl and benzyl azides and the N1-propargyl nucleobases previously synthesized. This new protocol allows the synthesis of a series of new 1,2,3-triazole derivatives with good to excellent yields (82–92%). The ADME (Absorption, Distribution, Metabolism, and Excretion) analysis showed good pharmacokinetic properties and no violations of Lipinsky’s rules, suggesting an appropriate drug likeness for these new compounds. Molecular docking simulations, conducted on different targets, revealed that two of these new hybrids could be potential ligands for viral and bacterial protein receptors such as human norovirus capsid protein, SARS-CoV-2 NSP13 helicase, and metallo-β-lactamase.