The property does not hold in general.

First, I’ll recall some basic properties of $\omega$-consistency. Let $T\vdash_1\phi$ denote the relation that $\phi$ is derivable from $T$ using rules of first-order logic, and *unnested* instances of the $\omega$-rule.

If $T\vdash_1\phi$, then $\phi$ is derivable from $T$ using a *single* instance of the $\omega$-rule. In particular, $T$ is $\omega$-inconsistent if and only if $T\vdash_1\bot$.

If $T$ is r.e., then $T\vdash_1\phi$ is a $\Sigma_3$ property of $\phi$. Thus, the $\omega$-consistency of $T$ is a $\Pi_3$ statement.

On the other hand, $Q\vdash_1\phi$ for every true $\Sigma_3$ sentence $\phi$: write $\phi=\exists x\,\forall y\,\psi(x,y)$, where $\psi\in\Sigma_1$, and fix $n\in\omega$ such that $\mathbb N\models\forall y\,\psi(\bar n,y)$. Then $Q\vdash\psi(\bar n,\bar m)$ for every $m$ by $\Sigma_1$-completeness of $Q$, hence $Q\vdash_1\forall y\,\psi(\bar n,y)$ using the $\omega$-rule.

Similarly, $Q+\mathrm{Th}_{\Pi_1}(\mathbb N)\vdash_1\phi$ for every true $\Sigma_4$ sentence $\phi$: this follows by the same argument as above, with $\psi\in\Sigma_2$.

If $T$ is an r.e. extension of $I\Delta_0+\mathit{EXP}$ (this can be negotiated down with a bit of care), then $T\vdash_1\phi$ satisfies the Bernays–Löb derivability conditions: that is, if we write $\Box_{T,1}\phi$ for the natural arithmetization of $T\vdash_1\phi$, we have

$T\vdash_1\phi\implies T\vdash_1\Box_{T,1}\phi$: this is a consequence of 2 and 3.

$T\vdash\Box_{T,1}(\phi\to\psi)\to(\Box_{T,1}\phi\to\Box_{T,1}\psi)$: we can concatenate two proofs.

$T\vdash\Box_{T,1}\phi\to\Box_{T,1}\Box_{T,1}\phi$; more generally, if $\psi\in\Sigma_3$, then $T\vdash\psi\to\Box_{T,1}\psi$. This follows by formalizing the argument in 3, using the ordinary formalized $\Sigma_1$-completeness of $Q$.

See [1] for more information about the provability logic of $\vdash_1$ and related provability predicates.

Now, let $T_0$ be an $\omega$-consistent r.e. extension of $I\Delta_0+\mathit{EXP}$ (such as $I\Sigma_1$ or $\mathit{PA}$), and put

$$T=T_0+\Box_{T_0,1}\bot$$

(that is, $T_0$ + its own formalized $\omega$-inconsistency). The standard proof of the second Gödel incompleteness theorem using the derivability conditions shows that $T$ is $\omega$-consistent. On the other hand, $\neg\Box_{T_0,1}\bot$ is a true $\Pi_3$ sentence, hence

$$T_0+\mathrm{Th}_{\Pi_1}(\mathbb N)\vdash_1\neg\Box_{T_0,1}\bot$$

by 4, thus $T+\mathrm{Th}_{\Pi_1}(\mathbb N)$ is $\omega$-inconsistent.

Kreisel’s lemma can be strengthened in a different direction, namely it holds for a larger class of formulas than $\Pi_1$:

**Proposition:** If $T\supseteq Q$ is $\omega$-consistent, and $\phi$ is a true $\Sigma_3$ sentence, then $T+\phi$ is $\omega$-consistent.

**Proof:** Otherwise $T+\phi\vdash_1\bot$, hence $T\vdash_1\neg\phi$. On the other hand, $Q\vdash_1\phi$ by property 3 above, hence $T\vdash_1\bot$, i.e., $T$ is $\omega$-inconsistent.

**Corollary:** If $T\supseteq I\Delta_0+\mathit{EXP}$ is $\omega$-consistent, and $A$ is an r.e. set of true $\Pi_2$ sentences, then $T+A$ is $\omega$-consistent.

**Proof:** It suffices to find a true $\Pi_2$ (or $\Sigma_3$) sentence $\phi$ such that $T+\phi\vdash A$. So, let $\alpha(x)$ be a $\Sigma_1$ definition of $A$ in $\mathbb N$, and $\mathrm{Tr}_{\Pi_2}(x)$ a universal $\Pi_2$ formula. Then we can take $\phi=\forall x\,(\alpha(x)\to\mathrm{Tr}_{\Pi_2}(x))$.

Notice that both statements are optimal with respect to arithmetic complexity:

The Proposition may fail for true $\Pi_3$ sentences $\phi$: for example, take the $\omega$-consistent theory $T=T_0+\Box_{T_0,1}\bot$ considered above, and $\phi=\neg\Box_{T_0,1}\bot$, which is a true $\Pi_3$ sentence. Then $T+\phi$ is inconsistent.

The Corollary may fail for r.e. sets $A$ of true $\Sigma_2$ sentences: continuing the previous example, write $\phi=\forall x\,\psi(x)$ with $\psi\in\Sigma_2$, and put $A=\{\psi(\bar n):n\in\omega\}$. Then $T+A\vdash_1\phi$, while $\neg\phi\in T$, hence $T+A$ is $\omega$-inconsistent.

**Reference:**

[1] George Boolos, *The Logic of Provability*, Cambridge University Press 1993.