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

or

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

or

I wish I could wake up early.