We develop an analytical framework to derive the meta distribution and moments of the conditional success probability (CSP), which is defined as {success probability for a given realization of the transmitters}, in large-scale co-channel uplink and downlink non-orthogonal multiple access (NOMA) networks with one NOMA cluster per cell. The moments of CSP translate to various network performance metrics such as the standard success or signal-to-interference ratio (SIR) coverage probability (which is the $1$-st moment), the mean local delay (which is the $-1$-st moment in a static network setting), and the meta distribution (which is the complementary cumulative distribution function of the conditional success probability and can be approximated by using the $1$-st and $2$-nd moments). For uplink NOMA, to make the framework tractable, we propose two point process models for the spatial locations of the interferers by utilizing the base station (BS)/user pair correlation function. We validate the proposed models by comparing the second moment measure of each model with that of the actual point process for the inter-cluster (or inter-cell) interferers obtained via simulations. For downlink NOMA, we derive closed-form solutions for the moments of the CSP, success (or coverage) probability, average local delay, and meta distribution for the users. As an application of the developed analytical framework, we use the closed-form expressions to optimize the power allocations for downlink NOMA users in order to maximize the success probability of a given NOMA user with and without latency constraints. Closed-form optimal solutions for the transmit powers are obtained for two-user NOMA scenario. We note that maximizing the success probability with latency constraints can significantly impact the optimal power solutions for low SIR thresholds and favour orthogonal multiple access (OMA).