Elaborating on Mark's comment, associated to a manifold of type CAT=PL,DIFF,TOP or a menagerie of others we have a disk bundle with structure group the CAT diffeomorphisms of the disk. In every case, we can interpret this disk bundle as a TOP disk bundle. By quotient by the sphere bundle (if we are over a compact space), we can form the Thom space. If our disk bundle is orientable with respect to a multiplicative cohomology theory, we get the cohomological Thom isomorphism.

Recall that to define Stiefel-Whitney classes for a vector bundle, we only use Steenrod operations and the mod 2 Thom isomorphism. Since every disk bundle is oriented mod 2, we can use the exact same definition. As well, if we have an integrally oriented disk bundle we can use the exact same definition of Euler class, as a pullback of the Thom class by the zero section.

An oriented CAT manifold is easily seen to have an integrally oriented disk bundle, so we have an Euler class. By picking a section of the disk bundle with isolated singularities (where it hits the zero section), we may mimic the proof in the DIFF case since the total index of the section of the disk bundle is the Euler characteristic of the manifold. This means the Euler class evaluates on the fundamental class to the Euler characteristic.

Then the same proof as the DIFF case shows that the CAT Stiefel-Whitney class is the mod 2 reduction of the CAT Euler class. And, again, the same proof as in the DIFF case shows that Stiefel-Whitney classes can be defined via your formula with the Wu classes, so this definition agrees with yours.

Elaborating on Mark's comment, associated to a manifold of type CAT=PL,DIFF,TOP or a menagerie of others we have a $\mathbb{R}^n$ bundle with structure group $CAT$. In all these cases, we can form the fiberwise one point compactification, and then identify these to get the Thom space. The normal proof of the Thom isomorphism goes through with the obvious notion of orientability of these bundles.

Recall that to define Stiefel-Whitney classes for a vector bundle, we only use Steenrod operations and the mod 2 Thom isomorphism. Since every disk bundle is oriented mod 2, we can use the exact same definition. As well, if we have an integrally oriented disk bundle we can use the exact same definition of Euler class, as a pullback of the Thom class by the zero section.

An oriented CAT manifold is easily seen to have an integrally oriented $\mathbb{R}^n$ bundle, so we have an Euler class. By picking a section of the disk bundle with isolated singularities (where it hits the zero section), we may mimic the proof in the DIFF case since the total index of the section of the disk bundle is the Euler characteristic of the manifold. This means the Euler class evaluates on the fundamental class to the Euler characteristic.

Then the same proof as the DIFF case shows that the CAT Stiefel-Whitney class is the mod 2 reduction of the CAT Euler class. And, again, the same proof as in the DIFF case shows that Stiefel-Whitney classes can be defined via your formula with the Wu classes, so this definition agrees with yours.

Elaborating on Mark's comment, associated to a manifold of type CAT=PL,DIFF,TOP or a menagerie of others we have a disk bundle with structure group the CAT diffeomorphisms of the disk. In every case, we can interpret this disk bundle as a TOP disk bundle. By quotient by the sphere bundle (if we are over a compact space), we can form the Thom space. If our disk bundle is orientable with respect to a multiplicative cohomology theory, we get the cohomological Thom isomorphism.

Recall that to define Stiefel-Whitney classes for a vector bundle, we only use Steenrod operations and the mod 2 Thom isomorphism. Since every disk bundle is oriented mod 2, we can use the exact same definition. As well, if we have an integrally oriented disk bundle we can use the exact same definition of Euler class, as a pullback of the Thom class by the zero section.

An oriented CAT manifold is easily seen to have an integrally oriented disk bundle, so we have an Euler class. By picking a section of the disk bundle with isolated singularities (where it hits the zero section), we may mimic the proof in the DIFF case since the total index of the section of the disk bundle is the Euler characteristic of the manifold. This means the Euler class evaluates on the fundamental class to the Euler characteristic.

Then the same proof as the DIFF case shows that the CAT Stiefel-Whitney class is the mod 2 reduction of the CAT Euler class.

Elaborating on Mark's comment, associated to a manifold of type CAT=PL,DIFF,TOP or a menagerie of others we have a disk bundle with structure group the CAT diffeomorphisms of the disk. In every case, we can interpret this disk bundle as a TOP disk bundle. By quotient by the sphere bundle (if we are over a compact space), we can form the Thom space. If our disk bundle is orientable with respect to a multiplicative cohomology theory, we get the cohomological Thom isomorphism.

Recall that to define Stiefel-Whitney classes for a vector bundle, we only use Steenrod operations and the mod 2 Thom isomorphism. Since every disk bundle is oriented mod 2, we can use the exact same definition. As well, if we have an integrally oriented disk bundle we can use the exact same definition of Euler class, as a pullback of the Thom class by the zero section.

An oriented CAT manifold is easily seen to have an integrally oriented disk bundle, so we have an Euler class. By picking a section of the disk bundle with isolated singularities (where it hits the zero section), we may mimic the proof in the DIFF case since the total index of the section of the disk bundle is the Euler characteristic of the manifold. This means the Euler class evaluates on the fundamental class to the Euler characteristic.

Then the same proof as the DIFF case shows that the CAT Stiefel-Whitney class is the mod 2 reduction of the CAT Euler class. And, again, the same proof as in the DIFF case shows that Stiefel-Whitney classes can be defined via your formula with the Wu classes, so this definition agrees with yours.