In to town today to buy milk and put fuel in the car, and I decided to stop in to the local motorcycle shop (Wimmera Motorcycle Co, principally an HD dealer but they sell lots of other shit as well) to see if they carry a coil for the BSA. I can order a 6V one off eBay for about 20GBP, but that'll be over $40AUD and then I'll have to wait for it to be shipped over here, and if there's anything wrong with it I'll have to ship it back.
I also want a 12V coil, because since I have to replace everything anyway I may as well do a 12V conversion and end up with something a lot more reliable. So I was prepared to pay a little bit more to be able to purchase it locally, the only caveat being that they have to order it, but it should be here sometime soon. It's a Lucas coil, which is probably neat from an originality perspective, but I really don't care too much about that and I'm not sure how I feel about Lucas electrical products.
It'll still need a battery and a regulator/rectifier (which I'm planning on going solid-state), but I'll buy those after I get the thing ticking over with a battery on the ground next to it.
Also asked about a rebuild kit for the Amal mono-bloc carb that's on it, but they don't carry that. They gave me another company in Melbourne that might be of use: Union Jack Motorcycles (03 9499 6428). I should probably also record the company that my great Uncle Jim recommended: Modak Motorcycles (03 9602 1229).