I got my invite yesterday (requested it the first day) and my delivery date is mid-May.
I'm pretty annoyed by this; even with slow production I think Amazon should have tried harder not to separate the sheep from the goats. They could have done it on a strict first-come, first-served basis. There's something about waiting on an invitation so I get to buy something from them (that other people have been enjoying for months and I tried to get it when they did) that is quite irritating.
I will probably not have Prime by the time this ships. I signed up in the waning days of the $79 fee and I'm going to let it lapse. I assume they'll still honor the Prime price.