There is simple infinite family of simple solutions with $m=n=1$.

The system is linear in $k_i$, giving the simple solutions:

$$ k_1 = (m^2*m' - 1)*p^{(-t + 2)}/q^t \\ k_2 = (m'*n^2 - 1)*q^{(-t + 2)}/p^t \\ k_3 = (m*m'*n - 1)*p^{(-t + 1)}*q^(-t + 1) $$

Solution is $m'=p^t q^t+1$.

There is simple infinite family of simple solutions with $m=n=1$.

The system is linear in $k_i$, giving the simple solutions:

$$ k_1 = (m^2*m' - 1)*p^{(-t + 2)}/q^t \\ k_2 = (m'*n^2 - 1)*q^{(-t + 2)}/p^t \\ k_3 = (m*m'*n - 1)*p^{(-t + 1)}*q^{(-t + 1)} $$

Solution is $m'=p^t q^t+1$.

For the edited question, solutions are $(m,n)=1+\mathbb{Z}p^tq^t,1+\mathbb{Z}p^tq^t$.

There is simple infinite family of simple solutions with $m=n=1$.

The system is linear in $k_i$, giving the simple solutions:

$$ k_1 = (m^2*m' - 1)*p^{(-t + 2)}/q^t \\ k_2 = (m'*n^2 - 1)*q^{(-t + 2)}/p^t \\ k_3 = (m*m'*n - 1)*p^{(-t + 1)}*q^(-t + 1) $$

For $m=n=1$ solution is $m'=p^t q^t+1$.

For the other case, fix small coprime $m,n$ and treat $m'$ as unknown. solution is $q^t \mid m^2*m' - 1, p^t \mid (m'*n^2 - 1)$ and then apply CRT.

There is simple infinite family of simple solutions with $m=n=1$.

The system is linear in $k_i$, giving the simple solutions:

$$ k_1 = (m^2*m' - 1)*p^{(-t + 2)}/q^t \\ k_2 = (m'*n^2 - 1)*q^{(-t + 2)}/p^t \\ k_3 = (m*m'*n - 1)*p^{(-t + 1)}*q^(-t + 1) $$

Solution is $m'=p^t q^t+1$.