Let $A<B$ and $(a_{n,m})_{n,m\geq 1}\subset \mathbb{R}$ a double sequence satisfying:
(1) For any fixed $m\geq 1$, the sequence $(a_{m,n})_{n\geq 1}$ is strictly increasing and $\lim_{n\rightarrow \infty}a_{m,n}=B$
(2) For any fixed $n\geq 1$, the sequence $(a_{m,n})_{m\geq 1}$ is strictly decreasing and $\lim_{m\rightarrow \infty}a_{m,n}=A$
Then, it is true that there is $k\geq 0$ such that
$\lim_{m\rightarrow \infty} a_{m,m+k}=\lim_{m\rightarrow \infty} a_{m,m+k+1}\neq A,B$
or
$\lim_{m\rightarrow \infty} a_{m+k,m}=\lim_{m\rightarrow \infty} a_{m+k+1,m}\neq A,B$ ?
Thanks in advances for you comments.
For $A < B$ and any real number $c > 0$ the double sequence $$ a_{m,n} = \frac{Bm + cAn}{m + cn} $$ satisfies the hypotheses, but $$ \lim_{m\to \infty} a_{m,m} = \frac{B+cA}{1+c} $$ can take any value in $(A, B)$.
Counter-examples to your updated question are $$ b_{m,n} = \frac{Be^m + An}{e^m + n} \, , \quad c_{m,n} = \frac{Bm + Ae^n}{m + e^n} $$ where $$ \lim_{m\to\infty} b_{m+k,m+l} = B \, , \quad \lim_{m\to\infty} c_{m+k,m+l} = A $$ for all $k, l \ge 0$.