When leaving to cut down a Christmas tree, Jack grabs an ax to cut down the tree. The cut on the tree they return with was cut smooth with a saw and it was not a fresh cut.
Jack took the three boys to find and cut down a Christmas tree with a double-bladed ax, but when they showed up afterward at the orphanage with a tree, it had clearly been cut down with a saw as the trunk had a smooth straight cut all the way through it with no ax marks on or around it.
Jack mentions the Beefeaters work at Buckingham Palace. They are the guards at the Tower of London, not Buckingham Palace.