That's an interesting question, Limpet!
If you are talking about slave trade pre-1950's (this is when she inherited the throne), I am of firm belief that what happened so far in the past is very much in the past, and we are not able to change what our ancestors did. Because of this I personally don't think it would make much difference if she apologized or not as she was not the one who made the decisions concerning such things - it was her ancestors. However, she could certainly apologize to those who are still living who were directly affected by the slave trade, though, as her ancestors are not living to do so.
On the same subject, other people's ancestors may have been affected by the slave trade as well, however their descendants are generally not experiencing the slave trade themselves, so while it may provide some peace of mind to their descendants to hear an apology, it doesn't change much in the way of the past.
I think the queen, or really any figure in a position of leadership over any country, would be better to apologize for any problems that have been created today as a result of the actions her ancestors took regarding slave trade. The leader can also back up the apology by trying to present solutions to fix the current problems - problems that might include direct or indirect racism, sexism, and job/pay/living disparities, as these are things she can actually take steps to fix or prevent.
All of this being said, however, if there were still a slave trade taking place now under her leadership, she would most certainly need to apologize for it and take steps to eliminate it.