Heavy metal pollution causes severe abiotic stress in cereal crops around the world. This study investigated the effects of different concentrations (0, 100, 200, and 300 mg·kg–1) of nickel, lead, and copper stress on the growth and biochemical responses of Aegilops tauschii seedlings, to provide a reference for research on the mechanism of invasion and screening potential sources of wheat tolerance genes. The results showed that nickel, lead, and copper stress caused a significant decrease in the contents of chlorophyll a, chlorophyll b, and chlorophyll (a + b) in A. tauschii, thereby inhibiting photosynthesis to different degrees and hindering seedling growth, which was reflected in significant reductions in plant height and root length, with the most notable effect observed under stress by 300 mg·kg–1 lead. As the concentration of heavy metals increased, the activities of antioxidant enzymes (SOD, POD, and APX), non-enzymatic antioxidants (GSH and AsA), and the contents of osmotic regulatory substances (proline and soluble proteins) in A. tauschii significantly increased. Additionally, heavy metal stress increased H2O2 and TBARS levels. However, when the nickel, lead, and copper concentrations reached 300 mg·kg–1, no significant differences were found in H2O2 or TBARS levels compared to those in the CK group. To summarize, A. tauschii can mitigate the accumulation of ROS and membrane lipid peroxidation caused by heavy metal stress through self-regulation, thus exhibiting a certain degree of tolerance to stress caused by different concentrations of nickel, lead, and copper. Finally, the evaluation using the membership function method revealed that among the three heavy metals, A. tauschii exhibited the strongest adaptation to Cu, followed by Ni and Pb.