BMW 318i 2004 Problems

If you've checked the battery and that's ok, check the starter to make sure there's no problem with that, and then pressure check for a blown head gasket. It's not unknown for a head gasket to blow on the 318 and you can get an hydraulic lock in a cylinder that will make it difficult to start if that cylinder is the one that's on compression when you go to start it. It doesn't always show up on the temperature gauge as overheating, but it is sometimes evident with white smoke from the tailpipe when the engine is cold.

There is a cooling-off period in Victoria. You have three clear business days from the time you sign the contract of sale to change your mind, but you automatically lose that right if you accept delivery of the car within that period. By taking it home your wife forfeited the right to the cooling-off period.