According to the NMM (and other sources) three flags from the Victory were carried at Nelson's funeral, the St. George's Ensign (i.e. the White Ensign) and two union jacks. The former would obviously have flown from the mizzen gaff and the two jacks were presumably those which flew from the fore and main topmast stays. (All the British ships present at Trafalgar had these hoisted to aid recognition, and which I believe were ordered by Nelson himself before the battle.)
That the NMM's fragment comes from the canton (or Union Jack part) of the ensign seems fairly evident, both from its size and the angle of the arms, but is it certain that this was the only flag torn by the seamen? I'm just speculating here, in that most authorities say that the fragments come from the ensign, or the 'largest flag', but is it just possible that some of the fragments may have come from either or both of the two jacks?
I have also been looking at Cowan's site again, at picture of the flag and the description beneath, and several things struck me. Firstly, and this may have been mentioned before, the flag itself appears too square and not of the right proportions and, secondly, I'm not so sure that the arms of the St Patrick's Cross would have come to a point where they join with the corners of the crosses of St George and St. Andrew. (Forgive me for saying so, but this is probably how a child would draw a Union Jack if asked!) I have done a bit of reading and most authorities say that the arms are cut off at an oblique angle, where they join with the white of the horizontal St Andrew's cross, (the angle depending of course on which side they are set) and neither do they run to the corners. (Look at any proper Union Jack and you will see that, due to the 'counterchange of the saltire' as it was called, some of the arms are set further back from the corners than others.)
Moving on to the description, there are several references here which point to to Cowan's not really knowing what they are talking about. Firstly, they seem to have assumed that Nelson was the senior admiral of the RN, not knowing that he was a 'mere' Vice-Admiral of the White, and therefore seem somewhat surprised that he was not flying a Red Ensign; John Cline according to them was the sailing master, but even if he was, he would not have been in charge of the seamen, but the navigation of the ship; then they say that the seamen kept a piece of the flag in rememberance of their dead captain. Presumably they knew that Nelson was an Admiral, but seem not to have known that he would have had a flag-captain to handle the Victory; finally, where did the Admiralty, in their letter, put forward the idea that Nelson's body was wrapped in the White Ensign before burial? It's the first I've heard of it. Did their Lordships really say that and, more to the point, were they actually taken in themselves?
Perhaps though, I protest (nitpick) too much?
