Bubba seems to be a relatively new thing - didn't happen when I was a kid or when my first nieces were born, but their younger siblings are referred to that way, often by the older sibling as well as parents.

As for beebo, are you sure it's not peepo? Peepo is the UK version of peek-a-boo, the game where you interact with a small child by blocking eye contact, then removing the blockage and making an exclamation. This can involve putting a blanket over the child, or just putting a hand in front of your face. In the UK that exclamation is "peepo!" and it sometimes gets used almost as a greeting to a small child in an excited kind of way. It's a bit like saying "boo!" but with with the fright aspect turned down.