To remove the transport term, consider the change of variables $v^\epsilon(t,x) := u^\epsilon(t,t+x)$. Notice that $v^\epsilon$ satisfies the heat equation $\partial_t v^\epsilon = \epsilon \partial^2_x v^\epsilon$. Thus $v^\epsilon(t,x)$ is just the convolution of $v$ with the standard heat kernel $\Phi(s,\cdot)$ at time $s=\epsilon t$, which is a function living at length scale $\sqrt{\epsilon t}$. I think you can then get an easy estimate for $v^\epsilon-v$ and translate it back into one for $u^\epsilon-u$. You will presumably need some info involving the smoothness of the initial data, unless you want your $C$ to have to depend on a lower bound for the time.

To remove the transport term, consider the change of variables $v^\epsilon(t,x) := u^\epsilon(t,t+x)$. Notice that $v^\epsilon$ satisfies the heat equation $\partial_t v^\epsilon = \epsilon \partial^2_x v^\epsilon$. Thus $v^\epsilon(t,x)$ is just the convolution of $v$ with the standard heat kernel $\Phi(s,\cdot)$ at time $s=\epsilon t$, which is a function living at length scale $\sqrt{\epsilon t}$. I think you can then get an easy estimate for $v^\epsilon-v$ and translate it back into one for $u^\epsilon-u$. You will presumably need some info involving the smoothness of the initial data.