BSA were very good at making incremental changes over the years, the result being that incorrect parts from different models will fit. The cylinder barrel is a case in point. Pre '53 engines have narrow fins, the fin size was increased when the Swing Arm framed models were introduced. The cylinder head you have there is the later wide fin alloy type. But the barrel appears to be the early type. Have a look at online images etc and the difference will jump out.
Internally the head bolt spacings, bore and combustion chambers are the same dimensions, so it all fits. But.....A7 and A10 barrels also differ in height and bore, likewise the cast iron and alloy heads A7 and A10 while externally similar in configuration, differ internally with respect to combustion chamber size, inlet tract and valves.
Luckily there is plenty of published data to identify what you have, most BSA castings are numbered, and parts books are available as reprints or conveniently in the Forum Literature Section. Draganfly also have an excellent comprehensive website. There is also Haynes, which is a good basic primer, but in retrospect, not without fault.
The oil pumps on these bikes are prone to stiffen up on long term storage, the body is alloy and prone to creep, and although the first instinct is to jump on the kicker, this can cause damage, so be warned.
Sourcing the correct barrel to match existing pistons is the ideal, but Sod's Law means heavy money to start with if you have to get new pistons, the Forum has lots of information as to the pitfalls of the various aftermarket brands.
Looking again at the picture at the start, the bodgers' art is epitomised by the long sticky out bolts on the front mudguard bridge, so be prepared for a few surprises along the way.
A note on nut and bolt threads.... Threads into alloy are generally Whitworth (BSW), most aftermarket fasteners are UNC which for most applications is fine, primary case screws for example. The rest of the nuts and bolts are CEI, also known as Cycle Thread or BSCy, but there are exceptions, Head Bolts being BSF. So, stripped and damaged threads are to be expected where some fool has forced an M6 bolt down a 1/4 Whit hole..... For replacements, Specialist suppliers, eg Nooky's Nuts in Essex are your best source, (as is Grandad's nut and bolt tin). Forum also has a Nut and Bolt Chart ( cross reference to part number) in the Literature Section. Not perfect, a few missing and unknowns, but overall pretty good.
Swarfy.