How can a person who woke up late while in a state of ritual impurity (junub), went to work without performing ghusl, and cannot perform ghusl at work, purify himself to pray Fajr, Dhuhr, and Asr when he returns home at Maghrib time?

What you did is not permissible, as it is not allowed to neglect prayer under the pretext of work. Prayer takes precedence over work. You must repent, perform ghusl, and make up all the prayers you missed. Do not repeat such an act unless you would be fired from your job and cannot find an alternative. In that case, it is permissible for you to perform tayammum and pray, but delaying the prayer is not permissible.