For my answer, I assume that compositions are on non-negative numbers (allowing only positive numbers does not change my statements, qualitatively). First, observe that $p(N,m,n)=\binom{m}{n}_{N+1}$, where $\binom{m}{n}_{N+1}$ is a generalized, or extended, binomial coefficient (so there you have an exact formula, not only a lower bound). Since $\binom{m}{n}_A\ge \binom{m}{n}_B$ if and only if $A\ge B$, a (very weak) lower bound for $p(N,m,n)$ is $\binom{m}{n}$, the ordinary binomial coefficient.
However, much better lower bounds can be obtained by noting that $\binom{m}{n}_{N+1}$ arises as the distribution of the sum of $m$ iid random variables, uniformly distributed on $\{0,\ldots,N\}$. Then apply the central limit theorem (or a variant of the DeMoivre-Laplace theorem) to get nice bounds ...