$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\om}{\omega}$Let $g:=f_0$. There is some real $\de>0$ such that
\begin{equation*}
\om(g,\de):=\max\{|g(y)-g(x)|\colon x,y\in[0,1], |y-x|\le\de\}<\ep/3<\ep/2.
\end{equation*}
Take now any $t_0,\dots,t_n$ such that $0=t_0<\dots<t_n=1$ and
\begin{equation*}
\de_k:=t_k-t_{k-1}<\de
\end{equation*}
for all $k\in[n]:=\{1,\dots,n\}$.

If $f\colon[0,1]\to\mathbb R$ is such that for all $k\in[n]$ and all $t\in[t_{k-1},t_k]$ we have $f(t)\in I_k:=[g(t_k)-\ep/2,g(t_k)+\ep/2]$, then $\|f-g\|<\ep$.

So, for any standard Wiener process $W$,
\begin{equation*}
w(g,\ep)=P(\|W-g\|<\ep)\ge p:=p_1\cdots p_n,
\end{equation*}
where
\begin{equation*}
p_k:=\min_{x\in J_{k-1}}P(x+W_t\in I_k\ \forall t\in[0,\de_k], x+W_{\de_k}\in J_k),
\end{equation*}
$J_0:=[0,0]=\{0\}$, $J_n:=I_n$, and $J_k$ is the closed interval that is the middle third of the interval $I_k\cap I_{k+1}$ for $k\in[n-1]$. Note that for $k\in[n-1]$ the length of the interval $I_k\cap I_{k+1}$ is $>\ep/2$ and hence the length of the interval $J_k$ is $>\ep/6>0$. Also, for each $k\in[n]$ the shortest distance from any point $x\in J_{k-1}$ to the set $\{g(t_k)-\ep/2,g(t_k)+\ep/2\}$ of the endpoints of the interval $I_k$ is $>\ep/6>0$.
So, $p_k>0$ for all $k\in[n]$, and hence $p>0$. Moreover, one can give an explicit expression of each $p_k$ in terms of $\de_k$, $I_k$, $J_k$, $J_{k-1}$ -- cf. e.g. Proposition 6.10.6, p. 533.

Thus, one can explicitly express the lower bound $p>0$ on $w(g,\ep)$ in terms of $\ep$, $t_1,\dots,t_n$, and $g(t_1),\dots,g(t_n)$.

For $g(t)=t (5 - 18 t + 12 t^2)$, $\ep=5/2$, $n=4$, $\de_k=1/4$ for $k\in[4]$, the picture below shows the graphs $\{(t,g(t))\colon t\in[0,1]\}$ (blue), $\{(t,g_-(t))\colon t\in[0,1]\}$ (gold), $\{(t,g_+(t))\colon t\in[0,1]\}$ (green), and a path of the Wiener process (gray) belonging to the event
\begin{equation*}
\begin{aligned}
B&:=\{g_-(t)<W_t<g_+(t)\ \forall t\in[0,1], W_{t_k}\in J_k\ \forall k\in\{0\}\cup[n]\} \\ &\subseteq\{\|W-g\|<\ep\},
\end{aligned}
\end{equation*}
where $g_\pm(t):=g(t_k)\pm\ep/2$ for $t\in(t_{k-1},t_k]$, $k\in[4]$. We have $P(\|W-g\|<\ep)\ge P(B)\ge p>0$.