Grammar: which one is it correct and why?

The first is past simple. You are telling a story about what happened to you earlier. You would use that if you were telling someone at school or work what happened to you that morning.

The second is present perfect. The event is still happening. You would use that speaking to someone at the bus stop or calling someone on the phone from the bus stop as the bus you missed goes away down the road.

Should actually be "This morning, I woke up very late and missed the bus" There's no need to bring up the subject of yourself more than once in a single sentence because you are already the subject.

It depends on how recently you missed the bus.

Probably the first one, though, because it sounds like you're explaining that you missed the bus because you woke up late. The second one sounds as though you missed the bus casually, you know, some time in the afternoon, and not because you woke up late.

This morning i woke up very late, so i missed the bus

The later, don't use contractions formally. Using 'have' is valid, but why, it just makes your sentence less coherent.

the first one.

either works