We can use wish + subject + past tense to express regret that a present situation is not how we want it:
I wish I had a car. = I don't have a car.
I wish I knew the answer. = I don't know the answer.
I wish I woke up early. = I don't wake up early.
We use wish + subject + would to express regret about an action that a third party is unwilling to perform.
I wish she would go home now. = I am sorry that she isn't willing to go home.
I wish he would get a haircut. = I am sorry that he refuses to get a haircut.
It would be strange to say:
I wish I would go home now
I wish I would get a haircut.
because I have the power to perform those actions if I want.
The subject of wish cannot be the same as the subject of would as this would be illogical. We cannot therefore have I wish + I would.
The example sentence is somewhat different because the speaker is not expressing regret over a unwillingness to wake up early but over an inability to do so. In this case, I wish I would wake up early seems acceptable to me. But I prefer:
I wish I woke up early
I wish I could wake up early.