$Z_{n,m} = \frac {(X_n +Y_m) -(n+m)}{\sqrt{X_n+Y_m}}$ converges to $N(0,1)$

Question

Given $X_n \sim P(n), Y_m \sim P(m)$ independent. How can I see that $Z_{n,m} = \frac {(X_n +Y_m) -(n+m)}{\sqrt{X_n+Y_m}}$ converges to $N(0,1)$ when $n,m$ go to inifinity? I know that $Z_{n,m} \sim P(n+m)$, but I don't see how can I apply the central limit theorem here. I saw a proof for the convergence of a similar distribution here Limiting distribution of $\frac {X_n -Y_m -(n-m)}{\sqrt{X_n+Y_m}}$ where $X_n,Y_m$ are independent Poisson but it involves characteristic functions. Is there a more straightforward way to do it in this case?