The observation of Pauli blocking of atomic spontaneous decay via direct measurements of the atomic population requires the use of long-lived atomic gases where quantum statistics, atom recoil and cooperative radiative processes are all relevant. We
develop a theoretical framework capable of simultaneously accounting for all these effects in a regime where prior theoretical approaches based on semi-classical non-interacting or interacting frozen atom approximations fail. We apply it to atoms in a single 2D pancake or arrays of pancakes featuring an effective $Lambda$ level structure (one excited and two degenerate ground states). We identify a parameter window in which a factor of two extension in the atomic lifetime clearly attributable to Pauli blocking should be experimentally observable in deeply degenerate gases with $sim 10^{3} $ atoms. Our predictions are supported by observation of a number-dependent excited state decay rate on the ${}^{1}rm{S_0}-{}^{3}rm{P_1}$ transition in $^{87}$Sr atoms.
