We travelled as a family of 4 (2 adults and 2 children). At the First Wing check in, small boxed BA100 Victoria sandwich cakes offered for the children. My wife and I would have liked one too but were too embarrassed to ask. Nothing special in GF lounge apart from the decorated horse.
Went to the gate, told flight was inoperable and hence advised to head back to the lounge. We did so, to then be told at the entrance to GF that the flight was ready for boarding after all and to head back to the gate (not before I spotted some more of the boxed cake at the entry desk, so got 1 each for me and my wife).
On board in ET, BA100 chocolate (hotel chocolat) handed out from a basket. Took my son to the loo and given another 4 pieces. As others have said, no mention of the big birthday at any time in any of the announcements or when the chocolate was handed out.
Overall, I thought the cake and chocolate were nice touches, although a free drink in ET would have been even nicer (as well as free pour LPGS in GF)