Whoever owns the car is financially responsible for any at-fault accidents that occur in it regardless of who the driver is. If the title is in your father's name that makes it him that is financially responsible. So if your car is uninsured one of two things may happen. If the other party had uninsured/underinsured motorist coverage than their insurance company will take care of their damage. Their insurance company will then come after the registered owner of your car, your father, and seek to collect what they had to spend.
If the other person didn't have uninsured/underinsured motorist coverage on their policy then they can sue the registered owner, your dad, for what it cost to fix their car. Whether they decide to take that route is of course up to them but it really depends just how much damage you caused and, especially, if there were any physical injuries as a result of the accident.
As for whether you can now buy car insurance to cover the accident the answer is an unequivocal no. You must have insurance at the time an accident occurs in order for it to be covered. There is no insurance company that will let you buy car insurance and then retroactively cover some accident that you already caused.