That still happens to me on a controller. On a wheel, never tried it.
From someone that has played every kind of kinky ass game, I can tell you this isn't so bad, just takes time to get used to.
The game uses a formula or smthin for the steering, like 0.33x in the first second, then full steering.

I have and I have never encountered artificial delays in steering. When you use a keyboard for the Simulator subgenre of racing games (which Unleashed is), then that isn't a steering delay that happens, it's just simulating your driver using the steering wheel. Your car still reacts immediately, it just won't immediately turn as sharply as it can like in arcade games.

When you switch to a controller or steering wheel that simulation disappears and it'll just steer the way you use your analog stick or wheel. In the case of this game here it doesn't matter whether you use a keyboard or controller, there's just a random delay on steering for me.

You've definitly never played any racing game that's been out since 2012 or newer.
They all have 2 second delays to "simulate steering". NFS Shift unleased 2 is the prime example of such garbage.
This one is tricky but playable.

In the case of this game it's both, but it really does have some of the worst handling systems I've encountered in a racing game to date. The cars barely react to button presses and when they do it's at a ridiculous delay.

"Car handling is... terrible"
Yeah, it's either that or you suck at playing racing games.
Wonder which one is it.