In overcast conditions natural light is scattered and diffuse, creating soft shadows. On bright sunny days the light is directional and harsh, creating hard-edged shadows. So, if you drop the sky from a bright sunny day into an image from an overcast day, the shadows could be a dead giveaway of fakery. The color and direction of the light in both the foreground and the dropped-in sky should match. Also, be aware of reflective objects in the foreground...they could indicate a fake sky.
Before dropping in a fake sky I'd first try to salvage the sky that you have, as it should look more believable than dropping in another sky. Is the sky completely blown out, or is there at least some detail? By decreasing exposure and increasing recovery I've been able to turn bland overcast skies into dramatic skies. If you can salvage the sky, use a layer mask to blend two copies of the same image -- one copy processed for the sky and the other copy processed for the ground.
If captured the image in RAW, you might be able to recover a lot of the sky. Post the image or provide a link to a large image.