I have a few brass whistles that are similar to the one on your web-site. I know the provenance of two, the third being from an antique dealer and all were obtained here in Australia. They are all cylidrical, made of brass tubing and have (or had) lead fipples.
One was passed on to me, along with some other material, by the daughter of Herb Gimbert (a traditional player who lived in Sydney in his later years) after he died. I have played it and found it sounds quite sweet. I understand that the heavy lead mouthpiece is an important factor in this ... but not too good for your health (despite which, Herb lived to 88). I placed a piece of plastic 'shrink-fit' tubing over the end and shrunk it into a tight fit, then removed sections to allow the whistle to play. This minimises the ingestion of lead.
Another came from another wonderful old traditional player, Sally Sloane (died aged ~84). I picked it up at her home in the late 1970s ... bent and missing its lead fipple after her grandchildren had played with it. She said I was welcome to it, so I set out to restore it.
I located a suitable brass rod and turned it down to a slight clearance fit in the tube. I used this as a mandrel to panel-beat out the dents and bends. I also ground the end of the rod to the shape of the fipple and straightened out the pressed whistle end that held the lead fipple.
I made a new fipple from brass, which I 'tinned' with lead-free food vessel solder to give the appearance of the original lead block. This was lightly fitted into place with epoxy resin, so that I could easily remove any non-authentic work. The whistle plays reasonably well but some notes are a little flattened because it has been played so much that the holes are slightly worn away.
The third whistle is very similar to the Herb Gimbert piece in pitch (a little sharp of the marked 'E'), sound and style, so I have not modified it in any way and do not play it.
All these whistles seem to be in "Old High Pitch", typical of their era. One of them (I need to look at them to remember which - probably Herb's) is branded "GENERATION" with letters stamped into the brass tube. I know that lead-fippled "Generations" were still available in parts of Australia after the Second World War (whenever they were actually made) and I knew players that looked far and wide for more of them. I imagine that some health authority eventually banned them from sale.
Sally Sloane's has a "Cigar Band" brand like yours, but is branded "Albert's Boomerang Brand" - the trademark of Jacques Albert, a Swiss luthier who became a very successful music dealer in Sydney and, in gratitude, used a great number of Australian names as trademarks - particularly 'Boomerang' which became the company name ... sa well as the name of the mansion he built.
I presume that some maker - or several makers (probably in England) would make these whistles for anyone ... and brand them as required with 'house brands' for music shops, or whatever was needed. Any more accurate identification would be a job for someone in the area in which they were made - possibly London, but more likely Birmingham or its environs.