We are privy to Emma's thoughts, and she never thinks that Harriet is anything other than the natural daughter of a gentleman - not a nobleman or royalty, just a gentleman. For this she has no evidence and is in fact completely wrong, but she is not so totally delusional as to think Harriet could have royal blood. The readers of JA's times would know that it was far from uncommon for men of almost all ranks, if they had the means and a sense of responsibility, to tuck their natural daughters into obscure boarding schools or worse. They would also know that royal bastards tended to remain with their mothers, and were either openly acknowledged (see the FitzClarences) or passed off as belonging to their mothers' husbands.