I recently wrote to the Anniversaries Office of Buckingham Palace to see if they will supply me with the data regarding total numbers of birthday cards sent out during 2021 (broken down by age and sex). This would enable us to provisionally identify gaps that may exist here (I'm fairly sure that you have to provide a birth certificate in order to claim a card, so I'd think false cases were unlikely). Whether they will supply the data is anybody's guess, but I tried to be as polite as I could in my letter and I can't really see it being a data breach (since I only want numeric data and no specifics about individuals). If not, then at least I got to feel temporarily regal by sending something to Buckingham Palace at the post office.

Of course, some people don't claim the cards so even if the data is supplied then there's no guarantee it's the exact total of people at each age, but I expect it's extremely rare that people don't get them so it'd be near enough complete.

@ell That's great to hear, I hope you get a reply soon. From what I have read, supplying a birth record isn't actually needed for a birthday card as they check the DWP Pensions list instead. I think the main discrepancy though will be that it's only optional to request a card, but I have no idea how many people don't request one.

@andrew I thought it was either sent automatically using DWP data (if you are registered receiving a pension), or you have to apply (in which case you need to prove your age somehow). In any case, I think false cases are unlikely (though probably not impossible). 

In virtually every article and post I've seen for people celebrating such milestone birthdays, it is mentioned that they received a card. I'd think that cases where people either don't claim one (or in Bob Weighton's case, opt out!) are very rare. 

I've also tried the DWP with a freedom of information request.