From Ising resonant fluctuations to static uniaxial order in antiferromagnetic and weakly superconducting CeCo(In$_{1-x}$Hg$_{x}$)$_{5}$ ($x$=0.01)

CeCo(In$_{0.990}$Hg$_{0.010}$)$_{5}$ is a charge doped variant of the $d$-wave CoCoIn$_{5}$ superconductor with coexistent antiferromagnetic and superconducting transitions occurring at T$_{N}$= 3.4 K and T$_{c}$=1.4 K, respectively. We use neutron diffraction and spectroscopy to show that the magnetic resonant fluctuations present in the parent superconducting phase are replaced by collinear $c$-axis magnetic order with three-dimensional Ising critical fluctuations. No low energy transverse spin fluctuations are observable in this doping-induced antiferromagnetic phase and the dynamic resonant spectral weight predominately shifts to the elastic channel. Static ($\tau$ $>$ 0.2 ns) collinear Ising order is proximate to superconductivity in CeCoIn$_{5}$ and is stabilized through hole doping with Hg.