I'll check out that CD. I should make it clear, if my opening post didn't, that I'm not asking whether English players play reels (i.e. reels from Ireland or Scotland) - they undoubtedly do - but whether there's such a thing as an audibly distinct, recognisable stylistic form as an English reel.

As someone pointed out above, loads of reels are Scottish in origin, but that didn't stop Ireland taking on the reel and making something that is unmistakably an Irish reel. This clearly hasn't happened in England on the same scale... but has it happened at all?

As others have pointed out above, the Northumbrian tradition is probably the most fruitful place to look. When I go on to YouTube, I can find several videos of musicians playing what Northumbrian reels. Here's one example:
https://www.youtube.com/watch?v=byEgZhPm1Sw

While I don't know or play that specific tune, I play several very like it. So it seems I play plenty of Northumbrian reels already (which I kind of suspected might be the case). Tunes like this, played this way, wouldn't (I'd have said) be regarded as reels in Irish music; they strike me as closer to an Irish polka or Irish march than an Irish reel.