Gödel’s second incompleteness theorem requires neither exponentiation nor “impredicative concepts”. The system Nelson works in (called $Q^*$ IIRC) is a fragment of arithmetic interpretable on a cut in $Q$, which includes at least the bounded arithmetic $I\Delta_0+\Omega_1$. The theory $I\Delta_0+\Omega_1$ (and even weak fragments of it with more restricted induction, such as $PV_1$) is perfectly capable of proving the second incompleteness theorem (for theories with a polynomial-time set of axioms, which is not a real constraint).

Gödel’s second incompleteness theorem requires neither exponentiation nor “impredicative concepts”. The systems Nelson works in are fragments of arithmetic interpretable on definable cuts in $Q$; one such fragment is the bounded arithmetic $I\Delta_0+\Omega_1$ (this appears to be what Nelson calls $Q_4$ in the Predicative arithmetic book). The theory $I\Delta_0+\Omega_1$ (and even weak fragments of it with more restricted induction, such as $PV_1$) is perfectly capable of proving the second incompleteness theorem (for theories with a polynomial-time set of axioms, which is not a real constraint).