Let $G$ be a group which is the (outer) semi-direct product of groups $N\rtimes H$. $n\in N,h\in H$. Is that true that the order of $(n,h)$ in $G$ is equal to the lcm of $\mathrm{ord}(n)$ and $\mathrm{ord}(h)$ ?
If it's not true, what may $\mathrm{ord}((n,h))$ be ? i. e. $\mathrm{ord}(n)\cdot\mathrm{ord}(h)$ ?
None of them is true. Take $G$ to be the dihedral group of order $2n$, so you can see it as the semi direct product of $N=\mathbb{Z}/n\mathbb{Z}$ and $H=\mathbb{Z}/2\mathbb{Z}$.
Take any non trivial rotation $\sigma$ and any symmetry $\tau$. Then the order of $\sigma\tau$ is $2$ (since it is again a symmetry).
If you really want the interpretation in terms of semi-direct product, take $(\bar{m},\tilde{1})$ , where $\bar{m}$ is not trivial.