I'm feeling a bit confused, for the lack of a better word (thanks, brain fog!). On the one hand, I can see where Lady Susan Hussey might have just been curious and maybe just chose her words poorly, but on the other I can also see where she might have been offensive.
I come from a culture where asking where someone is from is a very routine, natural thing - considering how vast my country is and how much diversity we have in terms of regions, religions, languages, customs etc. This is something that is reflected in countries like the UK where there's a massive influx of people from all places and cultures. And yes, we sometimes end up realising that the world is a much smaller place than we had thought, or we make it into a smaller place by forging connections. For example, it was only a couple of years ago that we realised that our family doctor's wife is our relative, since she belongs to the same village as my great-grandparents (could be blood relative, otherwise also we consider someone from the same village a relative - more likely than not that we share ancestors). How do we know? They shared something, we asked where she was from, and it went from there.
On the other hand, there can be a negative element to this curiosity, since many tend to discriminate based on those same parameters too.
But still our natural instinct is to ask and share these details, even to the point of being extremely intrusive at times (and now it's dangerous too, with the current obsession with being 'pure' Indians - doubt there are any with all the conquerors and settlers
).
Then I place myself, as a person of colour elsewhere, in that situation. Someone who has encountered racism before, or knows someone who has, or has just read the news at any point, would be quite wary and take such questioning as racist. Someone who's taking the interaction at face value might not. Also, someone who's been prepped to expect that a certain person (or in this case, an institution) might be racist would find racism in everything they said and did.
Basically, ultimately, it all boils down to intention which sets the tone, the words, and the personal bias (in the most neutral sense of the word) of the recipient. Message encoding and decoding - communication 101.
But also, I've met/observed IRL certain European diplomats who should have known better but still didn't even bother to hide their disdain for <insert racist descriptors for Indians/Asians>. So
The timing and the connections in this case though make the whole thing quite sus. Very likely that Lady Hussey didn't even say 'really from', which is usually misconstrued as the truly offensive part, but the BRF know that this won't go away and no defense would be accepted. Hence the rapid, and PR-wise smart, response. The damage is done though.