I don't recall Greener as an original mfg'r of the SMLE,,either the No1 or the No4 rifle.
I guess you can't discount anything being made up in the early post war years including sporterized SMLE's by Greener.
As far as the commercial Lee/Speed Sporters, BSA made most of them.
LSA also made some but they are not often seen.
They were made in several patterns from Officers Carbines of a more Military style configuration up to and including the fine Lee/Speed commercial sporter.
They were made mostly on the Lee Metford and Lee Enfield (Long Lee) actions. But BSA did offer some of the versions on the No1 rifle.
These were true commercial rifles, not x-military rifles rebuilt into sporters.
A very long list of makers bought and then marked and retailed those BSA products with their own name.
Retailers marks commonly are placed on the bbl or the dust cover of the Lee Sporters.
Here's a short list of some of the retailers names seen on the sporters including 2 versions of the BSA name:
ARMY & NAVY Co-operative Society Ltd.
Henry Atkin
Frederick Beesley
Bentley & Playfair
BIRMINGHAM SMALL ARMS Co. Ltd.
BSA & M. Co.
John Blanch & Son London
Thomas Bland & Sons
C. G. Bonehill
Charles Boswell
CHURCHILL (GUNMAKERS) Ltd.
Cogswell & Harrison Ltd.
COOMBES COMPANY Ltd.
W.W. Greener Ltd.
HOLLAND & HOLLAND, Limited
I. Hollis
Hollis, Bentley & Playfair
Holloway & Co.
Holloway & Naughton Ltd
W.J. JEFFERY & Co.
Charles Lancaster
Alexander Martin
Charles Osborne & Co. Ltd.
WILLIAM POWELL & SON
J. Rigby
J. Robertson
Victoria Small Arms Co.
WESTLEY RICHARDS & CO. LTD.
James Woodward
William Moore & Grey
Any true commercial Lee Sporter bolt rifle is a nice find.