Can nitrate and sulfate ions be oxidised during electrolysis of their molten compounds? $$\begin{align}
\ce{O2 + 4H+ + 4e- \;&amp;&lt;=&gt; 2H2O}\quad\quad &amp;&amp;E^\circ = +1.229\ \mathrm{V}\\
\ce{S2O8^2- + 2e- \;&amp;&lt;=&gt; 2SO4^2-}\quad\quad &amp;&amp;E^\circ = +2.01\ \mathrm{V}
\end{align}$$

From the above, I know that an aqueous solution of a sulfate salt undergoing electrolysis will not oxidise to peroxydisulfate ions over water being oxidised to oxygen at the anode unless it is at extremely high concentrations.

However, what if the solution is molten? Will the product at the anode be peroxydisulfate?

Also, can nitrate ions be oxidised? Perhaps to peroxynitrate or peroxynitrite?
Not only they can, but they must get oxidized. There is no other option. This is the only way the electric current can pass between the electrode and the solution.

Whether or not the products will be some peroxyanions is another question. At high temperatures typical of molten salts, I don't think they will survive. Hence I would expect $\ce{O2}$ + disulfate in the first case and $\ce{O2 + NO2}$ in the second.