Just note that angular velocity applies to all points on a rigid body and that a single point does not have angular velocity. Also, note that learning from Wikipedia is tricky because it is very inconsistent and sometimes erroneous.

Can you please explain what the figures represent. There are circles with points and arrows which appear tangential but could indicate out of plane. Where is $u$ and $v$ in the figures?

@ja72 Everything is happening in a flat plane. $C$ is rotating around $B$, $B$ is rotating around $A$. The arrows attached to $B$ and $C$ represent their velocity vectors and are $u$ and $v$ respectively.

The vectors $u$ and $v$ are described as rotational velocity in the question, but drawn as translational velocities in the diagram. Please clarify. Your question is still not clear to me. I want to help, but I don't know what exactly are you asking.

@ja72 My understanding of that Wikipedia article is that if we know the angular velocity of $A$ about $B$ and of $B$ about $C$, then we can determine the angular velocity of $A$ about $C$. In figures 1 and 2, the angular velocity of $B$ about $A$ and of $C$ about $B$ are both the same (in each case they are rotating clockwise with some given speed), and yet sure the angular velocity of $C$ about $A$ is different in each figure.

Per my first comment. There no such thing as angular velocity of point about another point. Angular velocity is a property of the entire reference frame or body, and where the center of rotation is does not affects its value. Also read this wikipedia section I authored to show the relationship between absolute and relative centers of rotation.

@ja72 I'm reading Physics For Mathematicians by Spivak in which he defines "the angular velocity of a particle" to be the cross-product $c\times v$ where $c$ is the position of the particle and $v$ its velocity. Is this concept usually known by a different name?

By definition a single particle has no spatial dimensions (it's a point of zero size) and hence rotation of a point is nonsensical. Rotation arises when a collection of particles (the body) move together such that their distances remain fixed. See Chasles theorem. Rotational velocity is the fixed vector $\vec{\omega}$ that is used to describe the individual velocity of each particle in a rigid body as $$\vec{v} = \vec{\omega} \times \vec{r}$$

@ja72 I always thought angular velocity was just a useful convention for representing a uniform circular motion. You take a vector parallel to the axis of rotation, and set its magnitude equal to the radians per second of the rotation. That's always how I understood it at school. Does that concept not exist at all?

You are taking about representing orbital motion with a rotation. That only works when the radial distance is constant, and hence the frame is rotating, and the one particle is riding along with the frame. We're still talking about the rotational velocity vector as defined above.

@Jack M I would like to discuss your question further if possible.

Hi there, thanks for your time

Ultimately what I'm trying to do is understand is this convention of representing uniform circular motion by a vector (direction gives the axis, magnitude gives the angular speed).

That convention always struck me as arbitrary (sure, it takes three free parameters to describe uniform circular motion in space, so you can pack them into a vector if you want, but why do that?)

So I was trying to understand where it comes from, and that addition formula on Wikipedia seemed like a good motivation

My understanding was that what that formula meant is that if the "vector representation of the circular motion of A around B" (I'm going to avoid any attempts at technical terminology because I'm clearly utterly confused about it) is added to the vector representation of the motion of B around C, you get the vector representation of the motion of A around C

but that's clearly wrong