What is the authenticity of the saying that it is Sunnah for a Muslim to attend to his nails, mustache, pubic hair, and armpit hair every forty nights, or whenever they grow long?

These practices are done whenever needed. It is preferable to do them every week, and they should not be left for more than forty days, as Anas (may Allah be pleased with him) said: "A time limit was set for us for trimming the mustache, clipping the nails, plucking the armpit hair, and shaving the pubic hair: that we should not leave them for more than forty nights." So, if they grow long before the forty-day mark, they should be removed. The considered duration depends on their length and the need for removal, and it varies according to individuals and circumstances.