Well, I believe that "The Throstles" may still be West Brom's "official" nickname. They adopted the name when they moved to The Hawthorns in 1900. However, "The Baggies" also dates back to around the same period, and was used as an unofficial nickname. It was around the 1980's that "The Baggies" seemed to become the most used, possibly because it sounded cooler!
Palace changed their nickname thanks to Malcolm Allison when he was manager. He changed their colours, and also the club crest. Instead of showing the original Crystal Palace, he used an Eagle holding a ball (based on the Benfica crest).