Here's one site I found for three speed Sturmey Archer specific maintenance, it goes through the whole disassembly process:

http://www.karrot.org/ascotto/three_speed/

Sheldon Brown's site also has some good info, and the shop he works for (Harris Cyclery) has a pretty good stock of SA parts. You can also get hubs on eBay pretty cheap if you poke around. That's how I got my spare. Of course, with that said, that hub doesn't look like any of the Sturmey Archer AW hubs I have seen.

I would leave the basket on... I doubt if it is original, but it s something many who would ride a 3 speed back in the day would add on. The one feature that is missing that probably would have been standard on most 3 speeds is fenders.

I am finishing up my Sears 10 speed, (now indexed 12 speed wide range) and I expect it to ride well. The reason I bring it up, is mine was built in Austria and has a very similar reflector on the fork. Mine is early 70's so that's where I would place your bike too... If the ub is a Sturmey Archer, they are pretty good at marking a year of manufacture on the hub.

Looks like a fun project!