  In this paper, our primary objective is to provide a fresh perspective on the
relationship between the $(\overline{N},p,q)$ method, which is a product of
relevant one-dimensional summability methods, and $P$-convergence for double
sequences. To accomplish this objective, we establish certain Tauberian
conditions that control the behavior of a double sequence in terms of both
$O_L$-oscillation and $O$-oscillation in certain senses, building a bridge
between $(\overline{N},p,q)$ summability and $P$-convergence, while imposing
certain restrictions on the weight sequences. As special circumstances of our
findings, we demonstrate that Landau-type $O_L$ condition with respect to
$(P_m)$ and $(Q_n),$ as well as Hardy-type $O$ condition with respect to
$(P_m)$ and $(Q_n),$ serve as Tauberian conditions for $(\overline{N},p,q)$
summability under particular additional conditions. Consequently, these results
encompass all classical Tauberian theorems, including conditions such as slow
decrease or slow oscillation in certain senses.
