Let $E$ be a normed $\mathbb R$-vector space, $\mu$ be a finite signed measure on $(E,\mathcal B(E))$ and $$\hat\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi}$$ denote the characteristic function of $\mu$.
Replying to a previous formulation of this question, Kavi Rama Murthy
has shown that if $E$ is complete and separable and $\mu$ is nonnegative, then $\hat\mu$ is uniformly continuous.
It is easy to see that his proof still works in the general case as long as we are assuming that $\mu$ is tight$^1$, i.e. $$\forall\varepsilon>0:\exists K\subseteq E\text{ compact}:|\mu|(K^c)<\varepsilon\tag1.$$
Taking a closer look at the proof, I've observed the following: Let $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the duality pairing between $E$ and $E'$ and $$p_x(\varphi):=|\langle x,\varphi\rangle|\;\;\;\text{for }\varphi\in E'$$ for $x\in E$. By definition, the weak* topology $\sigma(E',E)$ on $E'$ is the topology generated by the seminorm family $(p_x)_{x\in E}$.
Now, if $K\subseteq E$ is compact, $$p_K(\varphi):=\sup_{x\in K}p_x(\varphi)\;\;\;\text{for }\varphi\in E'$$ should be a seminorm on $E'$ as well. And if I'm not missing something, the topology generated by $(p_K:K\subseteq E\text{ is compact})$ is precisely the topology $\sigma_c(E',E)$ of compact convergence on $E'$.
What Kavi Rama Murthy
has shown is that, since $\mu$ is tight, for all $\varepsilon>0$, there is a compact $K\subseteq E$ and a $\delta>0$ with $$|\hat\mu(\varphi_1)-\hat\mu(\varphi_2)|<\varepsilon\;\;\;\text{for all }\varphi_1,\varphi_2\in E'\text{ with }p_K(\varphi_1-\varphi_2)<\delta\tag2.$$Question: Are we able to conclude that $\hat\mu$ is $\sigma_c(E',E)$-continuous?
EDIT:
In order to conclude that $\hat\mu$ is (uniformly) $\sigma_c(E',E)$-continuous, we need to that $(2)$ holds for $K$ replaced by an arbitrary compact $\tilde K\subseteq E$. Given $\varepsilon>0$, we can show $(2)$ by choosing the compact subset $K\subseteq E$ such that $$|\mu|(K^c)<\varepsilon\tag3.$$
We may then write \begin{equation}\begin{split}\left|\hat\mu(\varphi_1)-\hat\mu(\varphi_2)\right|&\le\underbrace{\int_{K\cap\tilde K}\left|e^{{\rm i}\varphi_1}-e^{{\rm i}\varphi_2}\right|{\rm d}\left|\mu\right|}_{<\:\varepsilon}\\&\;\;\;\;\;\;\;\;\;\;\;\;+\int_{K\cap\tilde K^c}\left|e^{{\rm i}\varphi_1}-e^{{\rm i}\varphi_2}\right|{\rm d}\left|\mu\right|\\&\;\;\;\;\;\;\;\;\;\;\;\;+\underbrace{\int_{K\cap\tilde K}\left|e^{{\rm i}\varphi_1}-e^{{\rm i}\varphi_2}\right|{\rm d}\left|\mu\right|}_{<\:2\varepsilon}\end{split}\tag4\end{equation} for all $\varphi_1,\varphi_2\in E'$ with $p_{\tilde K}(\varphi_1-\varphi_2)<\delta$, where $$\delta:=\frac\varepsilon{\left\|\mu\right\|},$$ but I have no idea how we can control the second integral.
EDIT 2
A "proof" of this claim can be (found in Linde's Probability in Banach Spaces), but I have no idea why this proof is correct, since he is concluding the continuity immediately from $(2)$ (for a single $K$):
Maybe we need to assume that $\mu$ is even Radon, i.e. that for all $B\in\mathcal (E)$, there is a compact $C\subseteq E$ with $C\subseteq B$ and $|\mu|(B\setminus C)<\varepsilon$. The author is actually imposing this assumption, but he obviously doesn't make use of it in his proof (he would need to consider an arbitrary compact $\tilde K\subseteq E$, as I did above).
$^1$ On a complete separable metric space, every finite signed measure is tight.
Best Answer
Partial answer: I will give a proof assuming that $E$ is separable. Of course this will give a proof when $E$ is not separable but $\mu$ has separable support.
It is an interesting fact that if support of $\mu$ exists in the sense that there is a smallest closed set of full measure then it is necessarily separable. [This requires Axiom of Choice]
Under this hypothesis it is known that $\mu$ is tight. Ref. Convergence of Probability Measures by Billingsley.
Let $\epsilon >0$ and choose a compact set $K$ such that $\mu (K^{c}) <\epsilon$. Then $$|\phi (x')-\phi (y')|$$ $$ \leq \int |e^{i \langle x', x \rangle}-e^{i \langle x', x \rangle}| d\mu (x)$$ $$\leq \int_K |e^{i \langle x', x \rangle}-e^{i \langle x', x \rangle}| d\mu (x)+2\epsilon.$$ So $$|\phi (x')-\phi (y')| \leq \|x'-y'\|\int_K \|x|| d\mu(x)+2\epsilon<3\epsilon$$ if $$\|x'-y'\| <\frac {\epsilon} {M\mu(E)}$$ where $$M=\sup \{\|x\|:x \in K\}$$.