I'm generalising here from your particular view on state apologies to a more general one about state responsibility. If that's not an accurate appraisal of your view, my apologies; I hope this post will at least spur you to clarify this point.
Disavowing the continuity of the State through time might, perhaps, be a defensible position, but let's be clear as to the ramifications.
To begin with, states shouldn't have to pay the debts incurred by previous governments: those debts were incurred by a totally different set of people, perhaps with wildly different motivations and ideologies.
If a government signs a treaty, then when they lose power, the next government is under no obligation to honour that treaty.
People do, generally, consider that governments must abide by the obligations of their predecessors. They may be wrong to do so, but an argument is surely required as to why we are all wrong when we think that.
An apology might seem like a minor thing, especially when the object of that apology is deceased. We should resist this view. An apology is an admission of guilt and an acknowledgement of moral responsibility: "We did that thing, and we were wrong to do it, and we are sorry for it." An apology is required because the British state, while having a different government and different citizens, is an entity with continuity through time that retains some element of responsibility for its actions in earlier ages. With time, and the change that happens over time, this responsibility may diminish or lapse entirely. But Alan Turing's death is not long in the past, and the spectre of homophobia still lurks within our society. An apology is past due.
From a utilitarian viewpoint, we might argue that apologies better allow people to "not hold those prejudices any more". By apologising for the mistreatment of gays, or ethnic minorities, or colonial subjects, a state sets out a position: this mistreatment is not acceptable. Such apologies endorse a particular social norm, giving it a weight it might not otherwise have had.
But in another I dont. Those things: treaties etc. are honours of state. BUT the prosecution of Turing was simply the application of law - a law adopted by an even earlier generation and inspired by societies homophobia.
In that sense the role of the government isnt quite the same.
If the aim of such an apology is the start of a wider work against homophobia then I am 100% behind it. But it isn't really - so it's just empty words.
Apologies are a way of acknowledging that what has happened was undesirable and wrong. It's also a way of signaling that you will try to prevent it from happening again. Taken from that perspective an apology is useful.
Of course, if the next step after an apology is "getting sued", there's not much incentive to ever acknowledge anything bad...
Well the apology doesn't come from the person for the person. It is from the State and the State has no predecessor, in a way. I know it sounds like a formality, in fact it is: it is a symbolic message with more than one meaning:
1. State can be wrong
2. As we could have be wrong in the past, we could be wrong now.
3. It is a good thing that we recognize that
4. People who are in a similar condition today, shall no fear anymore
5. If we ever try to do something similar, you can remind us of our apologies, at least.