Antimatter nuclei in cosmic rays (CR) represent a promising discovery channel for the indirect search of dark matter. We present astrophysical background calculations of CR antideuteron ($\overline{d}$) and antihelium ($\overline{He}$). These particles are produced by high-energy collisions of CR protons and nuclei with the gas nuclei of the interstellar medium. In our calculations, we also consider production and shock acceleration of antinuclei in the shells of supernova remnants (SNRs). The total flux of $\overline{d}$ and $\overline{He}$ particles is constrained using new AMS measurements on the boron/carbon (B/C) and antiproton/proton ($\bar{p}/p$) ratios. The two ratios leads to different antiparticle fluxes in the high-energy regime $E\gtrsim$ 10 GeV/n where, in particular, $\bar{p}/p$-driven calculations leads to a significanly larger antiparticle flux in comparison to predictions from conventional B/C-driven constraints. On the other hand, both approaches provide consistent results in the sub-GeV/n energy window, which is where dark matter induced signal may exceed the astrophysical background. In this region, the total antinuclei flux, from interaction in the insterstellar gas and inside SNRs, is tightly bounded by the data. Shock-acceleration of antiparticles in SNRs has a minor influence in the astrophysical background for dark matter searches.