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We study embeddings of the simple admissible affine vertex algebras$V_k(sl(2))$ and $V_k(osp(1,2))$, $k \notin {\Bbb Z}_{\ge 0}$, into the tensorproduct of rational Virasoro and $N=1$ Neveu-Schwarz vertex algebra withlattice vertex algebras.  We prove that the admissible affine vertex algebra $V_k(sl(2))$ can beembedded into vertex algebra $L^{Vir} (c_{p,p'}, 0) \otimes \Pi(0)$ where  $L^{Vir} (c_{p,p'}, 0) $ is suitable minimal Virasoro vertex algebra and$\Pi(0)$ is a vertex algebra of lattice type.  By using these realizations we construct a family of weight, logarithmic andWhittaker $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$--modules. As anapplication, we construct all irreducible degenerate Whittaker modules for$V_k(sl(2))$.

Realizations of simple affine vertex algebras and their modules: the cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$