"why shouldn't I say it's English, merely because there isn't an unbroken chain?"

The point that I was trying to make is that it is clear from your original post that you are not part of that unbroken chain. Neither am I for that matter. So if you've learned a style which is Irish then just because you're English doesn't turn it into an English style. And certainly not "traditional".

As musicians who've come to folk music we have to choose what we're going to play and how to play it. There are two approaches to this. One is the "purist" one - how was this music played by traditional fiddlers? The other is simply try to fit in with the other players in the session, who have a clear idea of what English fiddle should sound like (or perhaps not a clear idea, but they're sure that the way you play isn't it!)

Listen to some of the players Johnny Adams has recommended. Listen to the fiddlers in your local sessions. As with any instrument, try to find your own style which reflects and respects the tradition and yet allows you to express yourself.