There is no manual
as such for any pre wwII BSA.
Post WWII BSA did Service Sheets ( on Kim's CD
) that covered a single task then only around the 60's did they start making actual manuals
All of the tune up details and specifications were printed in the owers handbooks and that is the best you will get.
From time to time all of the handbooks were aglomerated into a single volume called "The Book of the BSA" which got published at randome times.
Each edition is a unique book so you have to get the one that covers your bike
So you are looking for edition 6 or 7 of "The book of the BSA" credited to "waysider" published by Pitman press.
BSA also did a 1930-1936 all model instruction book, covering all of the engines made obsolete by the new B & M range which was reissued in 1955 so there are a lot of them around.
There are a couple on evil bay Australia right now if you want one, 50 pages.
Mercian Motorcycle manuals
reproduced it reasonably well they might still have some copies.
The Motorcycle magazine also put out a reproduction, 1935 to 1940 but it is a bit light on for model W as it is mainly about the exciting new B & M engines.
Bruce Main Smith reproduced it very well ( proper litho print ) in 1973 so there are lot of these copies ouut there as well. These 2 are the same book, slightly altered layouts.
As for parts books. there are none.
BSA printed all models parts listings on a yearly basis but the pages just show each part so there is a page of heads and a page of barrels and a page of primary cases etc etc.
Good because you can track down more modern parts that will fit , or can be made to fit because BSA never changed parts numbers which is why in the middle of a page of 62- ???? part numbers you suddenly see a bolt or washer which is 12-1234.
Unfortunately with this lay out , you have no idea of what went where and which side the champher points to, all part of the fun.
Australia was BSA's primary export market itll 1950 so model W's will be a bit rare in the USA , although Canada being part of the Unon Jack waving throng probably got a few.