I LOVE these two songs and they share the same game however they are from two very different places. I do find it interesting that not only is the game the same but the topic is very similar.
"Biddy Biddy Hold Fast" is from Jamaica. I first learned it from Julie Swank at a ROCKE workshop. I use it to teach the diminution of syncopa.
"Queen Alexandra" is from Edinburgh and I learned it from Susan Brumfield. Susan's source is Golden City by James Ritchie. This is one of my favorite "ti" songs. It's also great for teaching 6/8 meter (which I do in 5th grade):
Game formation: circle or row of players with one student in the middle
Action: all students have their hands held out in front of them with their palms vertical, touching and almost closed. The one student in the middle also has his/her hands with palms touching, but with a toy ring between their hands. As the song is sung, the center player "slices" through the other players hands and drops the ring into some one's hand. The players then try to guess who has the ring at the end. If you play "Button You Must Wander" with your students, it's the same game except with a ring.
One thing that we do is while the guessing is going on, all students hum the song. If they want to make a guess, they raise their hands (still clasped together) and point to who they thing has the ring. The guessed player then opens their hands to reveal if the guesser is correct. To avoid the guessing from taking too long we limit either the number of guesses per round or the number of times we hum the melody. The person with the ring is the next center person.