Start out from likelihood function $L(\theta)$ (for a single observation) and make the reparametrization $\theta = h(\eta)$ to get the reparametrized log-likelihood $L(\eta) = L(h(\eta))$. Then the log-likelihood function of $\eta$ is $\ell(\eta) = \ell(h(\eta))$ so the Fisher information for a single observation is

\begin{align*} I_1(\eta) & = E_\theta\left[\left(\frac{d \ell(\eta)}{d\eta}\right)^2\right]\\ & = E_\theta\left[\left(\frac{d \ell(h(\eta))}{d h(\eta)}\frac{dh(\eta)}{d\eta}\right)^2\right]\\ & = E_\theta\left[\left(\frac{d \ell(\theta)}{d \theta}\right)^2\right]\left(\frac{dh(\eta)}{d\eta}\right)^2\\ & = I_1(\theta)\left(\frac{dh(\eta)}{d\eta}\right)^2. \end{align*}

Start out from likelihood function $L(\theta;Y_1)$ (for a single observation) and make the reparametrization $\theta = h(\eta)$ to get the reparametrized log-likelihood $L(\eta;Y_1) = L(h(\eta);Y_1)$. Then the log-likelihood function of $\eta$ is $\ell(\eta;Y_1) = \ell(h(\eta);Y_1)$ so the Fisher information for a single observation is

\begin{align*} I_1(\eta) & = E_\theta\left[\left(\frac{d \ell(\eta;Y_1)}{d\eta}\right)^2\right]\\ & = E_\theta\left[\left(\frac{d \ell(h(\eta;Y_1))}{d h(\eta)}\frac{dh(\eta)}{d\eta}\right)^2\right]\\ & = E_\theta\left[\left(\frac{d \ell(\theta;Y_1)}{d \theta}\right)^2\right]\left(\frac{dh(\eta)}{d\eta}\right)^2\\ & = I_1(\theta)\left(\frac{dh(\eta)}{d\eta}\right)^2. \end{align*}