This should not be hard.
WW and BSF sizes are the same but are described differently owing to the point that has been made about the diameter of the shank. All tools will be marked clearly, W, WW, BS or BSF. The size across the jaws is greater than the size marked on the spanner or wrench, because the size is determined by the diameter of the bolt on which the appropriate nut threads, as also stated already.
Forget the theory, think of the practice: in the sizes we use, 3/16ths Whitworth is 1/4 BSF - or Cycle for that matter, as it follows the same spanner sizing even though the thread angles are different. It's the bolt heads and the nuts we are interested in. 1/4 W is 5/16ths BSF, 5/16ths W is 3/8ths BSF etc. 1/2 BSF is therefore 7/16ths W, going the other way. WW is always 'bigger' because with the coarse thread of the WW form the nut that goes on it is the same size across the flats (AF) as the nut that would go on a finer threaded - ie BSF - shank one imperial size larger. Needs to be too, if you think about the amount of metal left outboard of the coarse thread in a relatively small nut. For Cycle, the BSF sizing is the one to follow. On which point it is worth noting that although many people say there ain't any real difference between 1/4 BSF and 1/4 Cycle (that'll take a 3/16th WW or 1/4 BSF spanner, to illustrate the point) - both being 26 threads per inch and, natch, a 1/4 diameter - there is. In most cases it isn't really all that important - but if it's a serious bit of engine you want to hold together, it matters a lot. The Cycle thread has a 60 degree included angle (like UNC and UNF - the 'American' sizes), and like Metric. All BSF and WW thread forms, following the design of the good Mr Whitworth, are 55 degrees. Almost every tool I have ever seen, and I have seen a lot and own a lot, are clearly marked in both WW and BSF sizes, as in '3/16ths WW, 1/4 BSF'. A few cheap modern imports from funny places only state their dimensions in WW (I have some of those too), but if you know the basic principle you won't be fooled. If you're wiser than me you won't waste hard-earned on too many weapons from funny places - they don't compare in terms of ergonomics, let alone finish and quality.
What we need to fettle our toys are 6 basic sizes- 3/16ths, 1/4, 5/16, 3/8, 7/16 and 1/2 in. WW (aka 1/4 to 9/16th BSF) open-ended, ring, combination, sockets or any mix thereof. (Even old-fashioned cheapo box spanners, which have many advantages - 'thinth' and length being the 2 most important). I prefer rings, but they can't always be got onto the fastener. Go one size up at the top end if available. For things like clutches, drive sprockets, brakes (as mentioned) a few larger sizes come in handy and can be got when needed. If you can buy 3/8ths-drive sockets in the smaller sizes (to 5/16ths WW - those are the most important sizes anyway) they are more versatile and quite strong enough - Britool and KingDick make them still), but 1/2 drive will do for almost everything.
BA, mentioned in the above thread, is a valuable series I couldn't live without. Usually found on electrical baggage like magnetos and dynamos etc, it's a metric-based system (0BA is 6mm dia by 1mm pitch for example) but the thread angle is 47.5 for all BA sizes. For practical purposes, the sizes we need are 0BA, 2BA, 4BA and sometimes 6BA.The weapons are available still, in open-ended, ring, or socket form. Sometimes, nothing else will do, although 4BA and 1/4AF are for practical purpose interchangeable and will get a contact breaker assembly out of a Lucas magneto. It's a beautiful thread form and the cost of a few proper weapons far exceeds the hassle of rounding fasteners with 'ignition spanners' of dubious provenance and banging small metric sockets onto nuts and bolts not designed for them.
Now, if you're serious about wanting the right gear to do stuff right, you need to think about threading tackle too - to reclaim all the things other people have mangled and to make anew the things that you can't reclaim . . . a workshop without taps and dies (all easily obtainable in the sizes we need) is only a garden shed. But hey, the anorak factor has to be curtailed . . .
Groily