Varia radar for running?

It would detect them. It wouldn't work through a watch unless you were recording a bike activity. You could run an RTL515 through your phone while you record a run on your watch though.

Data said:

And could it also warn for (fat)bikes and scooters or are those too small?

As a user of Varia RTL 516 (515 for German market), I can confirm that not only cars are detected reliably, but also motorcycles and I also had it once or twice, that it detected another biker on a fast ride (and probably 20 years younger than me ;) ) and gave me a warning.

BUT

Data said:

Would a Varia radar work if you clip it on a belt and wear it on your back when running, or would the motion of running throw it off?

I seriously doubt, that it would work with all the motion that you have, when just clipping it to your back and then go running.

On a bike, you have to watch out that the radar axis is parallel to the ground, directly backwards and not impacted by a wheel or a (saddle-)bag (not mounted to low or to high). And then it is quite static.
On your back while running I guess because of the motion of your hip and back it would constantly change angle by 20, 30 degrees or even more.
I don't think it will work like this, reliably.

Of course you could use some kind of a gimbal and mount the Varia to that and then wear the gimbal, but I seriously would be interested to see that construction then ;)

Indeed, that would make an interesting contraption! :-) I still like to experiment with a unit just strapped to my back, but I'll start with something cheap (and probably nasty) from Aliexpress.