We can rehearse the arguments for and against till the cows come home (in fact, we've already done so, haven't we, Mark?

) but I do think the final proof lies in the intimate letters that Nelson wrote to Emma when he revealed his innermost thoughts about their relationship. It was risky to put anything of the sort down on paper as their mail was intercepted, hence the Thompson subterfuge; but when he knows he is 'safe' because letters were transmitted personally by trusted hands, he drops all pretence of the Thompson involvement and makes it clear that Horatia is 'our little child'. I could quote several, but the one beginning, 'My dearest wife, for such you are in and the eyes of heaven,' is particularly telling. That sentence he wrote after the birth of Horatia, when he rejoiced effusively in fatherhood and referred to their daughter as the 'pledge of love' that Emma had given him is, I think, quite conclusive. There can be no other inference, I think, than that Emma was Horatia's mother.
As for the Thompson stories - they are riddled with inconsistencies: Thompson 'served in Victory with Nelson' - though I believe no Thompson is on the muster roll; he was the 'son of a sailmaker'; he served in the Elephant and was killed at Copenhagen etc. etc. The story of Thompson as Horatia's father was supported and propagated by Nelson's brother officers, Hardy and Berry in particular. Officers were always prepared to provide 'cover' for each other to disguise their extra-marital escapades. ('Every man is a bachelor after Gibraltar.')