Classically, the dual under the Seiberg-Witten map of noncommutative U(N), = 1 super Yang-Mills theory is a field theory with ordinary gauge symmetry whose fields carry, however, a θ-deformed nonlinear realisation of the = 1 supersymmetry algebra in four dimensions. For the latter theory we work out at one-loop and first order in the noncommutative parameter matrix θ^μν the UV divergent part of its effective action in the background-field gauge, and, for N≠1, we show that for finite values of N the gauge sector fails to be renormalisable; however, in the large N limit the full theory is renormalisable, in keeping with the expectations raised by the quantum behaviour of the theory's noncommutative classical dual. We also obtain –for N ≥ 3, the case with N = 2 being trivial– the UV divergent part of the effective action of the SU(N) noncommutative theory in the enveloping-algebra formalism that is obtained from the previous ordinary U(N) theory by removing the U(1) degrees of freedom. This noncommutative SU(N) theory is also renormalisable.