Is it permissible for an engineer to take financial compensation from the trust fund for using his private car in the project work, given that the company did not provide the promised car and that not using his car would delay the work?

The basic principle is that it is not permissible to take anything from a trust that one is entrusted with. Your taking from it constitutes treachery and consumption of unlawful (haram) wealth. Using your private car for work, if it is a requirement of your employment contract, then it is binding upon you. Otherwise, it is not binding, and you have the right to refuse until the employer provides a vehicle. Using the car for work with the intention of taking compensation from the trust without knowledge [of the employer] is a betrayal and an aggression that is not permissible. So, beware of risking your faith and your integrity with such flimsy interpretations.