What is the ruling on someone who had intercourse with his wife after Fajr prayer and then slept, waking up late for work, and was unable to perform ghusl and prayer until returning home at night? Is his prayer valid, and what should he have done?

You have committed a great error and a grave sin by delaying the ritual bath (ghusl (definition)) and missing prayers. Delaying prayer beyond its prescribed time is one of the major sins, and ablution (wudu') does not suffice for the ritual bath (ghusl) from Janabah (definition) (state of ritual impurity). You should have sought permission from work to state your excuse. Continuing to work out of shyness before people, while Allah is more deserving of shyness, exposed you to Allah's wrath. You must now perform sincere repentance and make up the prayers you offered without purification.