has just had or just had a baby

She has just had a baby is absolutely idiomatic in British English.

I observe that She just had a baby seems to be idiomatic in American English (and has been creeping into British use over the last few decades); I think she has just is also idiomatic in American usage, but I'm not so sure of that.

Your understanding of the perfect is not quite right (unsurprising, as it's very subtle). You're right that it means the speaker is thinking of the action as having present relevance; but it does not imply that she is still having it. It's more that the just brings the event so close to the present. I would hardly ever use the simple past with just - in my idiolect they are inconsistent.

[Of course there is a different meaning of just which I do use with the simple past, just meaning something like only, as in I just touched it and it fell!.]

And by the way, there is no such thing as "the official grammar".