The variance recurrence
Derive \operatorname{Var}(Z_{n+1})=m^2\operatorname{Var}(Z_n)+\sigma^2 m^n from the law of total variance, and check that the critical-case formula is the limit of the m\neq 1 formula as m\to 1.
Conditional on Z_n, the next generation is a sum of Z_n independent offspring, so its conditional mean is m Z_n and its conditional variance is \sigma^2 Z_n. The law of total variance gives
\operatorname{Var}(Z_{n+1})=\mathbb{E}(\sigma^2 Z_n)+\operatorname{Var}(m Z_n)=\sigma^2 m^n+m^2\operatorname{Var}(Z_n),
which is the recurrence v_{n+1}=\sigma^2 m^n+m^2 v_n. At m=1 it reduces to v_{n+1}=v_n+\sigma^2 with v_0=0, so v_n=n\sigma^2. For the m\neq 1 formula, (m^n-1)/(m-1)\to n as m\to 1, and \sigma^2 m^{n-1}\to\sigma^2, so the product tends to n\sigma^2.