I did exactly this last year - call up; 10% of cash fare (or Avios, if the adults are already on an award booking: no need for separate award availability for baby, unless you are trying to get them their own seat).
Do check in MMB before the day of travel that the e-ticket number has come through for the baby - mine did not, and a quick phone call solved this (and probably saved a lot of faff at the airport).
Turn up a little early to allow some extra process time at the airport the first time round - though we had no delay or problem at all when checking in with her for the first time.
We were dreading the packing/checking in/security process with a baby, but in fact if you do your research quickly and know the rules it's pretty simple and nothing to be worried about: granted, we have only done LCY and First Wing with her, but the process re: prams, milk (liquids) and the rest is the same for everyone. The BA website has a good section on travel with kids which should answer many of your questions.
Overall, I have been pretty impressed by BA's handling of all things baby-related: have not found the travel with her especially stressful. I suspect that's about to change as she is now old enough not to just want to sleep on planes, but that's another story....
Congratulations!