The eleven new copper(II) and nickel(II) coordination compounds [Cu(L)Br]2 (1), [Cu(L)Cl] (2), [Cu(L)NO3] (3), [Ni(L)Cl] (4), [Ni(HL)2](NO3)2 (5), and [Cu(A)(L)]NO3, where A is 1,10-phenanthroline (6), 2,2′-bipyridine (7), 3,4-dimethylpyridine (8), 3-methylpyridine (9), pyridine (10) and imidazole (11) were synthesized with 3-(morpholin-4-yl)propane-2,3-dione 4-allylthiosemicarbazone (HL). The new thiosemicarbazone was characterized by NMR and FTIR spectroscopy. All the coordination compounds were characterized by elemental analysis and FTIR spectroscopy. Also, the crystal structures of HL and complexes 1, 6, 7, and 11 were determined using single-crystal X-ray diffraction analysis. Complex 1 has a dimeric molecular structure with two bromide bridging ligands, while 6, 7, and 11 are ionic compounds and comprise monomeric complex cations. The studied complexes manifest antibacterial and antifungal activities and also have an antiradical activity that, in many cases, surpasses the activity of trolox, which is used as a standard antioxidant in medicine. Copper complexes 1–3 have very weak antiradical properties (IC50 > 100 µM), but nickel complexes 4–5 are strong antiradicals with IC50 values lower than that of trolox. The mixed ligand copper complexes with additional ligand of N-heteroaromatic base are superior to complexes without these additional ligands. They are 1.4–5 times more active than trolox.