The lap infant always gets 1x hand luggage and 1x 23kg, regardless of class of booking or status of the travelling parents. I think an infant gets 1x 23kg even if the parents are on HBO fares, which is a neat trick for a long weekend away...

You do also get a free checked pushchair (gate-checked) and car seat with an infant.

I have never really needed more than this just for the little one, though in fairness with the Gold allowance we each get when we travel we would just add extras under our names.

Infant policy on BA is actually reasonably generous. One area where (so far) I have always found ground staff and crew more than helpful.