Just racked my brain over this and in fact what I thought was Reckless which was closing down on my last visit was in fact called Rhythm Records and it was in the very same shop location as Honest Jons. You are correct, it did have a spiral staircase and the jazz was downstairs.
I must have been there around 2010/2011. I thought it looked strangely familiar - in fact I recognised the old wooden racking from the HJ days. I came out with a couple of A&M Horizons from their sadly very depleted stock (hardly anything left). I wouldn’t be surprised if these were old stock from the 1970s ! All very sad though - what used to be reggae in the main section upstairs had morphed into the usual non-musical tourist tat and most of the space downstairs was taken up with cheap t-shirts etc.
Bill - worth getting hold of the Garth Cartwright book as there is quite a bit on Dobell’s, Rays, Mole etc. in there. The ‘Bermuda Triangle of the West End’, ie. Oxford St/New Row/Shaftesbury Ave up to King’s Cross !
Definitely the same place.