Heat the oil in a large, heavy-based frying pan and cook the potatoes over a medium-high heat, turning frequently, for 7–10 minutes until golden and crisp.
Meanwhile, heat the olive oil in a separate large frying pan and cook the cod loins over a high heat for 3–5 minutes, turning once, until golden and cooked through. Squeeze the lemon juice over the fish and season with salt and pepper.
Mix the lemon rind and half the dill into the mayonnaise and serve with the fish and chips. Scatter the remaining dill over the fish to serve.