  We consider $C^*$-algebra $\mathcal{E}_{n,m}^q$, which is a $q$-twist of two
Cuntz-Toeplitz algebras. For the case $|q|<1$ we give an explicit formula,
which untwists the $q$-deformation, thus showing that the isomorphism class of
$\mathcal{E}_{n,m}^q$ does not depend of $q$. For the case $|q|=1$ we give an
explicit description of all ideals in $\mathcal{E}_{n,m}^q$. In particular
$\mathcal{E}_{n,m}^q$ contains unique largest ideal $\mathcal{M}_q$. Then we
identify $\mathcal{E}_{n,m}^q / \mathcal{M}_q$ with the Rieffel deformation of
$\mathcal{O}_n \otimes \mathcal{O}_m$ and use a K-theoretical argument to show
that the isomorphism class does not depend on $q$.
