The nuclear liquid-gas transition at normal nuclear densities, $n \sim n_0 = 0.16$ fm$^{-3}$, and small temperatures, $T \sim 20$~MeV, has a large influence on analytic properties of the QCD grand-canonical thermodynamic potential. A classical van der Waals equation is used to determine these unexpected features due to dense cold matter qualitatively. The existence of the nuclear matter critical point results in thermodynamic branch points, which are located at complex chemical potential values, for $T > T_c \simeq 20$ MeV, and exhibit a moderate model dependence up to rather large temperatures $T \lesssim 100$ MeV. The behavior at higher temperatures is studied using the van der Waals hadron resonance gas (vdW-HRG) model. The baryon-baryon interactions have a decisive influence on the QCD thermodynamics close to $\mu_B = 0$. In particular, nuclear matter singularities limit the radius of convergence $r_{\mu_B/T}$ of the Taylor expansion in $\mu_B/T$, with $r_{\mu_B/T} \sim 2-3$ values at $T \sim 140-170$ MeV obtained in the vdW-HRG model.