As an Australian, I had never heard of Buxton Lawn, but just did some reading. It's an interesting story, but how do you prove it, unless you do some DNA testing? The articles I read have no mention of illegitimate children.. In fact his wife was almost constantly pregnant! The British royals weren't backward about acknowledging their illegitimate children. Throughout British history, most of them have been given titles eg The FitzRoys (descendants of Charles II) and their descendants still occupy the stately homes they were awarded. It seems strange that, if Buxton was a royal scion, that he was not acknowledged in any way. As I said, do some DNA testing and get it compared to the royal DNA. But remember: it might answer the question, but it will not entitle you to anything. There are millions of people with royal descent, and illegitimacy has traditionally barred them from inheriting the throne or receiving anything financial from the Royal Family.