Indeed it is. The faded logo is "John Bull" the folk hero to (some parts of) the UK, hence the Union Jack (or Union Flag for the purists) Waistcoat. Usually only good for one inner tube repair....next time round you'd already used the patch you needed and the skinflint quantity of rubber solution (glue sniffer's favourite) had dried in the tube to a solid lump.
Tony. Spanner or wrench sizes will be marked BSF or Whitworth. These two threads have common "across the flats" sizes. BSA use Whitworth threads (nice and coarse) into alloy castings, nuts and bolts are generally BSCY or "Cycle Thread" reflecting BSA's heritage and being a fine thread suitable for applications where vibration can lead to fixings becoming loose. BSF /Whit spanners fit fine. For sure, Metric and AF sizes can be left in the toolbox, they cause more problems with rounded off fasteners as they just don't fit well enough.
The literature section contains a nut and bolt chart thanks to Musky. It is a cross reference to fastener type from published part numbers. Covers most applications.
The adjustable spanner is also known as a Girder Wrench, a popular brand was "King Dick" which always raised a smile amongst the folks familiar with Limey Slang.
My parts book 1954-1957 (Printed April 1958) details a toolkit for Plunger and Swinging Arm Bikes, Page 79. Illustration Plate 33
If that's the bike pictured in your intro, please add a few drops of the blackest used oil you can find to the paving to add that authentic BSA touch.. On second thoughts, look in the oiltank. If it's full, you don't need us. If it's empty, you certainly do.
Welcome along.
Swarfy.