Let $X_{0}\sim N(0,1)$, $X_{1}$ another RV independent of $X_{0}$. Let $X_{t}=(1-t)X_{0}+tX_{1}.$ We know that as $t\to1,$ $\mathbb{E}[X_{0}|X_{t}]\to\mathbb{E}[X_{0}|X_{1}]=E[X_{0}]=0.$ Can we determine the rate of this convergence? For instance, if $X_{1}\sim N(0,\sigma^{2}),$ we can compute everything explicitly so that $\mathbb{E}[X_{0}|X_{t}]=\frac{1-t}{(1-t)^{2}+\sigma^{2}t^{2}}$ and hence $\frac{1}{1-t}(\mathbb{E}[X_{0}|X_{t}]-\mathbb{E}[X_{0}|X_{1}])\to\frac{1}{\sigma^{2}}$ as $t\to1.$
The question more precisely is, are there some mild assumptions on $X_{1}$ (besides $X_0\perp X_1$) so that $\frac{1}{1-t}(\mathbb{E}[X_{0}|X_{t}]-\mathbb{E}[X_{0}|X_{1}])$ converges to something finite?
Comments: I tried taking derivative wrt time of $\mathbb{E}[X_{0}|X_{t}]$ but it gets very messy.