Congratulations on the Bat Mitzvah.
I am not sure about how BA would accommodate the situation, however, with the catering reduction, I would not expect the Do&Co facility to prepare a cake upon request, etc...
Although I find BA crew very helpful in these kinds of situations and if you manage to bring a cake in good shape to the aircraft ( or a small piece of cake bought from an airport vendor ), as a goodwill gesture, the crew would be able to serve you that.
By the way, no one has mentioned it but if you're going to stay in NY state, there is a quarantine for 14 days which applies for passengers arriving from several countries including the United Kingdom.