In uniform circular motion, $\displaystyle \omega$ ,

**the angular velocity by definition is constant irrespective of the radius.**
Think of a spinning disc. The number of revolutions/sec made by each point on the disc is the same. Thus if the disc makes 10 revolutions per sec, so does each point. Thus each point makes an angular displacement of $\displaystyle 2\pi . 10$ or $\displaystyle 20\pi$ per sec and thus has the same angular velocity.

However, the linear velocity of each point on the disc is different. As r increases away from the centre, v increases accordingly to maintain $\displaystyle \omega$ constant.

When you use

and say

there is the implicit assumption that v is constant which in fact it is not as it has r dependence.

However when you use $\displaystyle F = m\omega^2 r$ , both m and $\displaystyle \omega$ are constant and hence

which is perfectly OK