Is it permissible to use my mother's company car and private driver to drop me off for part of my journey, which coincides with theirs?

The use of a company car is subject to the company's regulations and what has been agreed upon, as the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions." So, if the company owners or its authorized manager grant permission, there is no harm. The general rule is that it is not permissible to use company property for personal purposes without permission. Exceptions include what does not change or get damaged by use, and what is customarily overlooked. This includes a person riding with the driver in the company car.