How could Zefram Cochrane have made a warp drive engine if dilithium is needed to make one...

You'll recall that his warp drive was a crude prototype, and only went for a very short flight. It would not have been noticed at all, had the warp signature not been detected by Vulcans flying past the Solar System at the time. The dilithium-powered warp drives were later models introduced by other alien races who did business with Humans during the centuries after First Contact.

Remember, the Enterprise's engines are actually powered by antimatter, the dilithium crystals aren't a power source, but are used to either channel or stabilize the energy from the antimatter.

So if Cochran was somehow using raw antimatter to power his ship, without delirium, it's a miracle that he didn't blow himself to subatomic particles.