Here's a sample aircraft shot taken in conditions of overcast skies and drizzle. Viz was bad enough that the next flight was cancelled and the whole show packed in for the day. The exposure here was set manually at 1/200, f/7.1, 400 ISO and is unedited apart from white balance. There is no doubt some editing would improve it, but this is just to show an example of an exposure under the conditions you describe. It wouldn't hurt for it to be a little brighter, and I think maybe f/5.6 would have been a better choice than f/7.1.
I chose 1/200 because I wanted to capture some blur in the prop for other photos taken during the flight. For non prop planes I would usually be aiming for higher shutter speeds as there is no prop blur required. For drizzly conditions I suggest you start off at more like 1/400, f/5.6, 800 ISO and adjust from there. Of course there is cloudy and there is cloudy, so one size does not fit all, but that's a starting point which ought at least to get you a visible image. You can fine tune from there.
In conditions of bright sunshine you could be looking to shoot at maybe 1/800, f/5.6, 100 ISO or 1/800, f/8, 200 ISO. There are many ways to slice and dice the problem. Between bright sunshine and a drizzly day you are looking at around a four stop difference in the ambient brightness. There are many combinations of exposure setting which could be used to cover that range. There is no one right answer to any of it. You have to consider your own limitations, that of your equipment, the speed of the subject and also your creative intent for the control of motion and DOF, as well as deciding how you want the image exposed.
That may sound like a lot to get your head around, but it will fall into place. You should also get used to checking the histogram to verify exposure accuracy, as viewing the preview image on the camera alone may mislead you. Note I included the histogram in my screen capture above. That's important! It shows me I have a large gap on the right hand side, which, with the benefit of hindsight, is why I think I should have pushed the exposure higher in camera.
