In this paper, an effective fully-discrete implicit scheme for solving linear reaction-diffusion equations is constructed by using the variable-time-step two-step backward differentiation formula (VSBDF2) in time combining with the nonconforming finite element methods in space. By introducing a modified energy projection operator, a discrete Laplace operator, the discrete orthogonal convolution kernels, we obtain the optimal and sharp error estimates of order O(h2+τ2)\documentclass[12pt]{minimal}
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\begin{document}$$O(h^2+\tau ^2)$$\end{document} in L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}-norm and O(h+τ2)\documentclass[12pt]{minimal}
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\begin{document}$$O(h+\tau ^2)$$\end{document} in H1\documentclass[12pt]{minimal}
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\begin{document}$$H^1$$\end{document}-norm under a mild restriction 0<rk<rmax≈4.8645\documentclass[12pt]{minimal}
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\begin{document}$$0<r_k< r_{\max }\approx 4.8645$$\end{document} for the ratio of adjacent time steps rk\documentclass[12pt]{minimal}
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\begin{document}$$r_k$$\end{document}. Furthermore, with the help of a modified discrete Grönwall inequality and the combination technique of interpolation and projection operators, we achieved the superclose result between the interpolation function Ihu\documentclass[12pt]{minimal}
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\begin{document}$$I_hu$$\end{document} and finite element solution uh\documentclass[12pt]{minimal}
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\begin{document}$$u_h$$\end{document} in H1\documentclass[12pt]{minimal}
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\begin{document}$$H^1$$\end{document}-norm of order O(h2+τ2)\documentclass[12pt]{minimal}
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\begin{document}$$O(h^2+\tau ^2)$$\end{document}, which together with the interpolation postprocessing operator Π2h\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _{2h}$$\end{document} leads to the global superconvergence result about u-Π2huh\documentclass[12pt]{minimal}
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\begin{document}$$u-\Pi _{2h}u_h$$\end{document} in H1\documentclass[12pt]{minimal}
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\begin{document}$$H^1$$\end{document}-norm of order O(h2+τ2)\documentclass[12pt]{minimal}
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\begin{document}$$O(h^2+\tau ^2)$$\end{document}. Finally, numerical tests are provided to verify the theoretical analysis.
