Calculate the number of iterations in unusual nested loop
If you want to compute an upper bound of the asymptotic complexity, you can consider the maximum value for the most inside sum. Hence,$\sum_{n=n_0}^{n_T}1 \leqslant n_T + 1$. Therefore, we will have:

$$
N \leqslant \sum_{j=0}^{j_T}\sum_{k=0}^j\sum_{l=0}^k (n_T+1) = \sum_{j=0}^{j_T}\sum_{k=0}^j (n_T+1) (k+1) = (n_T+1) \sum_{j=0}^{j_T}\sum_{k=0}^j (k+1)  = 
$$
$$
(n_T+1)\sum_{j=0}^{j_T}\frac{(j+1)(j+2)}{2} = \Theta(n_T (j_T)^3)
$$

Therefore, $N = O(n_T (j_T)^3)$.