Nonlinear vibration and instability of embedded double-walled carbon nanocones subjected to axial load are investigated in this article based on Eringen's nonlocal theory and Timoshenko beam model. The elastic medium is simulated as Pasternak foundation and the van der Waals forces between the inner and the outer layers of double-walled carbon nanocones are taken into account. Using von Kármán geometric nonlinearity, energy method and Hamilton’s principle, the nonlocal nonlinear motion equations are obtained. The differential quadrature method is applied to discretize the motion equations, which are then solved to obtain the nonlinear frequency and critical fluid velocity of viscous-fluid-conveying double-walled carbon nanocones. A detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, thickness-to-length ratio, temperature change, apex angles, elastic medium and van der Waals forces on the dimensionless frequency and critical buckling load of double-walled carbon nanocones. The results show that the small-size effect on the nonlinear frequency is significant and cannot be neglected; also, the nonlinear frequency and critical buckling load decrease with increasing the cone apex-angle.