The answer is michaeltibbsjr, who, in the past 24 hours, has posted photos of both on Show & Tell.
The cookie jar appears to be a piece of Shawnee Pottery, which made lots of pig-shaped cookie jars, and even more than one version of Winnie the Pig. As for the Nuclear Weapons Warning Sign, we’re happy to take michaeltibbsjr at his word regarding the story he tells about that. Welcome to CollectorsWeekly.com!
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I also have the same cookie jar that was my grandmothers. It is marked with USA on the bottom. I have a set of pig salt and pepper shakers with blue around the neck that I thought was a set to the cookie jar, but I guess not since they are a different color.