Plenty of recordings of real English traditional singers are available, but as a rule you have to seek out specialised record labels for them; they are produced and issued for love, not money. I don't know who the 'academics with an agenda' Frank refers to are, but they always seem to get blamed for everything that can't be put down to the mysterious 'folk police' anyway.

Of course, many of the recorded Revival performers one hears are at least semi-pro; their performances need to be polished if they are to make any sort of a living. The old singers were, in the main, just ordinary people with day jobs who happened to know some old songs, and comparing them on the same terms is fairly pointless; though some were indeed musicians of some stature. To Cox, Larner and Taylor I'd add Walter Pardon, Fred Jordan and Phil Tanner; but there were many more. There were also some remarkable stylists such as Henry Burstow, of whom no recordings survive. There is a surviving cylinder recording of David Clements singing 'The Banks of Green Willow', though, which can be heard (in two instalments) at the British Library's Collect Britain website:

EFDSS Cylinder No.97 and EFDSS Cylinder No.104

The collector George Gardiner was extremely impressed with Mr Clements' singing. He was about 80 when recorded; just think how he would have sounded in his prime. Martin Carthy was no less impressed, and it is Mr Clements' version of the song that he sings.

That nasal style wasn't much used, so far as we can tell, by English traditional singers. A lot of Travellers use it, though, and the early Revival seems to have picked it up from them. It is much more common in Ireland. The 'folk club' style is, essentially, a modern construct made of bits and pieces from all over the place. It has been reinforced by the tendency of younger performers to get their repertoire from records made by other revivalists, though of course there are honourable exceptions.

Concertina accompaniment for shanties may sound 'quintessentially English' to some people (and perhaps it is, now) but it's another product of the Revival rather than the tradition. It goes back about 50 years.