I once asked Imin Chen this very question around 13 years ago, and he pointed me to the 1992 paper "Sur une question de B. Mazur" by Kraus and Oesterlé. They work out an explicit example for $N = 7$. Imin also gave me examples for other $N \geq 8$:

$$\begin{aligned}
N & = 11: & &
\begin{aligned} E: & y^2 = x^3 + x^2 + 2 x + 2 \newline E': & y^2 + x y = x^3 + x^2 + 81710302x+ 576603336052 \end{aligned}
\newline \newline
N & = 11: & &
\begin{aligned} E: & y^2 = x^3 - 3x - 34 \newline E': & y^2 = x^3 - 362988 x + 82933524 \end{aligned}
\newline \newline
N & = 11: & &
\begin{aligned} E: & y^2 = x^3 - 27x + 918 \newline E': & y^2 = x^3 - 40332 x - 3071612 \end{aligned}
\newline \newline
N & = 13: & &
\begin{aligned} E: & y^2 = x^3 + x - 10 \newline E': & y^2 = x^3 - 362249x + 165197113 \end{aligned}
\end{aligned}$$

(I've never verified these claims myself.) Apparently the 1996 paper "Sur la comparaison galoisienne des points de torsion des courbes elliptiques" by Halberstadt and Kraus discusses what happens for composite $N$. As for what the Jordan Ellenberg mentioned, there is the 1998 paper "Modular diagonal quotient surfaces" by Kani and Schanz. (I didn't know David did this in his thesis. He was on my thesis defense committee, and we talked about mod 5 representations quite a bit!)

As for the other cases of $N \leq 6$, You may wish to take a look at the paper by Alice Silverberg entitled "Explicit families of elliptic curves with prescribed mod $N$ representations" in the volume "Modular forms and Fermat's last theorem". She works out formulas for $N = 3, 4, 5$. The papers "Mod 2 representations of elliptic curves" and "Mod 6 representations of elliptic curves" by Rubin and Silverberg round out the other cases.