A classical way of tackling such kind of problems is via Poisson approximations. For example, consider a Poisson point process in $(0,1) \times (0,\infty)$ with unit intensity. The number $N_k(T)$ of points in $(\frac{k}{P},\frac{k+1}{P}) \times (0,T)$ is distributed as a Poisson random variable with mean $\frac{T}{P}$: this represents an approximation of the number of beans in the $k$-th box. Indeed, the advantage of this Poissonization is that the random variables $N_k(T)$ are independent. The probability that at time $T$ each box contains at least $N$ beans is thus given by $\big(\mathbb{P}[\text{Poiss}(T/P) \geq N] \big)^P$, and you can then do all kind of asymptotic estimates.

I doubt that you will find a very tractable answer to your original question. Are you interested in the limit $N,P \to \infty$ ?

A classical way of tackling such kind of problems is via Poisson approximations. For example, consider a Poisson point process in $(0,1) \times (0,\infty)$ with unit intensity. The number $N_k(T)$ of points in $(\frac{k}{P},\frac{k+1}{P}) \times (0,T)$ is distributed as a Poisson random variable with mean $\frac{T}{P}$: this represents an approximation of the number of beans in the $k$-th box at time $T$. Indeed, the advantage of this Poissonization is that the random variables $N_k(T)$ are now independent - this was not the case in the original problem. The probability that at time $T$ each box contains at least $N$ beans is thus given by $\big(\mathbb{P}[\text{Poiss}(T/P) \geq N] \big)^P$, and you can then do all kind of asymptotic estimates.

I doubt that you will find a very tractable answer to your original question. Are you interested in the limit $N,P \to \infty$ ?