Can nitrate and sulfate ions be oxidised during electrolysis of their molten compounds?
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.