Is the mentioned ghusl, with its cleansing of the private parts, pouring water over the head, then washing with shampoo and soap while ensuring water reaches the nose and sides, then washing the feet and performing wudu, considered a valid ghusl?

The obligatory part of ritual bathing (ghusl) after major ritual impurity (janabah) is to spread pure water over the entire outer surface of the body with the intention [of performing ghusl]. If you cleanse your body with pure water and then use soap or shampoo, or vice-versa, your ghusl is valid. However, performing ghusl solely with water mixed with soap or shampoo is not sufficient according to the majority of scholars, though some scholars have validated it.