  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.
