[MrJohz]

What British style guides? The BBC doesn't mention it (and indeed uses it in their style guide). I don't have the full Oxford style guide, but the online version I've found uses singular they. It's obviously not British specifically, but interesting from the perspective of non-native speakers: the EU style guide also recommends using singular they (and again uses it in their style guide). The GDS is obviously pro singular they!

So I'm not sure where you're getting the idea that British style guides do not support this construction.

I don't think there's a particular benefit to using it, other than that it is a very common and suddenly understood form of English, and in many cases easier to understand than an equivalent, but more complex construction. You talk about using it in a formal context, which to a certain extent this is, but more than that it is important for Government websites to be widely understood - if the Government adopts a register that is too far removed from common English, then they will not be as easily understood. Given that there is no apparent confusion with singular they in conventional speech (and given that, in cases where it is ambiguous, it is possible to find alternative sentence constructions), it seems pointless not to use it.