As I recall, McAloon's method for proving that there are no computable nonstandard models of $I\Delta_0$ was to show that there are initial segments that are nonstandard models of PA. The usual Tennenbaum tricks can then be used to show that addition and multiplication are not computable.
Additional Comment-- Here are references for McAloon's paper and the paper of Wilmers that proves a similar result for $IE_1$ the fragment of $I\Delta_0$ where you only have induction for formulas with bounded existential quantifiers
McAloon, Kenneth, On the complexity of models of arithmetic. J. Symbolic Logic 47 (1982), no. 2, 403--415.
Wilmers, George Bounded existential induction. J. Symbolic Logic 50 (1985), no. 1, 72--90.