CuSO4/hydrazine hydrate was used as a catalyst system for copper(I)-catalyzed alkyne-azide cycloaddition (CuAAC) of AZT and 5′-azido adenosine with terminal alkynes to give 30 novel 1,2,3-triazole derivatives. Screening for their anticancer, anti-inflammatory, angiotensin-converting enzyme 2 (ACE2), and 3C-like protease (3CLpro) inhibitory activities showed that several triazoles of AZT containing murayafoline A and indirubin-3′-oxime inhibited the growth of HepG2 and LU-1 with the IC50 values ranging from 11.01 to 19.87 μg/mL. Besides that, some triazole derivatives of adenosine exhibited anti-inflammatory activity against RAW264.7 cells with the IC50 values within an interval of 12.00–59.48.00 μg/mL. Especially, two triazoles of adenosine with indirubin-3′-oxime at O- and N1 positions expressed the ACE2 and 3CLpro inhibitory activities in which the triazole of adenosine with indirubin-3′-oxime at N1 inhibited both ACE2 and 3CLpro inhibitory activities with IC50 values of 135.62 and 142.95 μg/mL, respectively.