$\newcommand{\si}{\sigma}
\newcommand{\ep}{\varepsilon}
$
Without loss of generality, $\ep\ge4/n$, because otherwise the bound $Ke^{-n\ep^2}$ on a probability (with an unspecified constant $K$) is trivial.
Let $S_m:=\sum_1^m X_i$. Then for $t\ge0$
\begin{equation}
S_{N_{nt}}\le nt<S_{N_{nt}+1}.
\end{equation}
If the event $\sup_{0\le t\le T}|\frac{N_{nt}}n-\frac t\mu|>\ep$ occurs, then for some $t\in[0,T]$ we have
\begin{equation}
\text{ either (i) $N_{nt}>k:=\lfloor n(\tfrac t\mu+\ep)\rfloor$ or (ii) $N_{nt}+1<l:=\lceil n(\tfrac t\mu-\ep)\rceil+1$ },
\end{equation}
whence either
$$S_k\le S_{N_{nt}}\le nt\le(k+1-n\ep)\mu\le(k-n\ep/2)\mu$$
or
$$S_l\ge S_{N_{nt}+1}\ge nt\ge(l-2+n\ep)\mu\ge(l+n\ep/2)\mu;$$
note also that here
\begin{equation}
k\le N:=\lfloor n(\tfrac T\mu+\ep)\rfloor, \quad
l\le\lfloor n(\tfrac T\mu-\ep)+2\rfloor\le\lfloor n\tfrac T\mu\rfloor\le N.
\end{equation}
So,
\begin{equation}
P \left( \sup_{0 \leq t \leq T} \left| \dfrac{N_{nt}}{n} - \dfrac{t}{\mu} \right| \geq \ep \right)
\le Q_1+Q_2,
\end{equation}
where
\begin{equation}
Q_1:=P(\min_{k\le N}T_k\le-n\ep\mu/2),\quad
Q_2:=P(\max_{l\le N}T_l\ge n\ep\mu/2),
\end{equation}
\begin{equation}
T_k:=\sum_1^k Y_i,\quad Y_i:=X_i-\mu,
\end{equation}
so that $EY_i=0$.
For positive $h$ close enough to $0$, we have $Ee^{h|Y_i|}<2$, whence
\begin{equation}
Ee^{hY_i}\le E(1+hY_i+h^2Y_i^2e^{h|Y_i|}/2)\le1+h^2\si^2\le e^{h^2\si^2},
\end{equation}
where $\si^2:=EY_i^2=Var\,X_i$.
Using now Doob's maximal inequality for the submartingale $(e^{hT_k})_{k\ge1}$, with small enough $\ep$ and
\begin{equation}
h=\frac nN\,\frac{\ep\mu}{4\si^2}
\end{equation}
we have
\begin{multline}
Q_2\le\exp\{-hn\ep\mu/2\}Ee^{hT_N} \\
\le\exp\{-hn\ep\mu/2+Nh^2\si^2\}
=\exp\Big\{-\frac{n^2\ep^2\mu^2}{16\si^2 N}\Big\} \\
\le\exp\Big\{-\frac{n\ep^2\mu^2}{16\si^2(T/\mu+\ep)}\Big\}
\le e^{-cn\ep^2},
\end{multline}
where $c:=\mu^3/(32\si^2 T)$.
Similarly, $Q_1\le e^{-cn\ep^2}$. So, the probability in question is upper-bounded by $2e^{-cn\ep^2}$, for small enough $\ep$ (depending on the distribution of $X_1$ and $T$). This bound is similar to what you wanted, $Ke^{-n\ep^2}$, except that the factor in the exponent has the extra constant factor $c$, depending on $\mu,\si,T$ -- as it of course should be.
