Hello everyone,
Sturmey Archer AW claims another newbie victim - me.

Had to take the rear wheel off (I keep a stock of inner tubes because it happens quite often), and obviously had to disconnect the adjustment cable. Since I only actually fixed the shifter not too long ago, this was a new ordeal and hence I wasn't expecting it to be that much of an issue.

Now, I followed/tried to follow Sheldon's and other advice, regarding visual and slack adjustment. I manage to get reliable 3 gear(HI) and 1(LO) with 3->2 engaging neutral and 1->2 working most of the time before slipping. Trying to remedy by tensioning the cable I eventually get reliable 3, neutral 2 and not enough cable slack to even shift down to 1. With the latter, I can sometimes wiggle the cable/adjuster assembly and free up some cable slack to shift down.

So, what could be the issue? The cable assembly is all original, apart from the adjuster being new, so it could be fairly worn, but it was working before. Visually, when I shift 3->2 the cable moves very slightly and 2->1 is much more visible. I oiled the adjuster chain to ease its movement out of the hexnut but that didn't seem to help much.
Another issue is that the shifting cable itself is quite tense by default, as in when I screw it onto the adjuster I have to apply considerable force, and unless I hold it or lock it with the nut it will just unscrew itself at some point. I can also only get the adjuster spindle about halfway in the barrel, it won't really go any further. Barrel threading issue?
Apologies for the wall of text, just wanted to provide enough info.
Any suggestions appreciated!