were, would, past simple...What comes after "I wish".

Introduction.

You can say
I wish I were a bird, I would fly to you.
But you can't say
I wish I would be a bird, I would fly to you.

be (am, are, is) don't take "would". simply uses "were".
normal verb takes "would". but not always.

For example.

incorrect: I wish I would have wings.
correct: I wish I had wings.

incorrect: I wish I would know the sky.
correct: I wish I knew the sky.

on the one hand.

correct: I wish I would fly to you.

what is different about?
This is different about "something to happen" and "something to be".

"something to be" means the state verb. have, know, like, love, believe...
"something to be" means the action verb. play, fly, run, swim... most of verb other than state verb.

conclusion

What comes after "I wish".
be (am, are, is) : were
state verb: past simple form of the verb
action verb: "would" + infinitive form of the verb