Don't be afraid to adjust your old Sturmey Archer three speed hub. It is a simple job that involves not much more than tightening or loosening the tiny 'indicator' chain that goes into the axle.
Note that this page primarily applies to the 'classic' AW hub that was made from the 1930s to the 1990s. This hub has a neutral positon between high and second gears. The instructions on this page also apply to other olde Sturmey Archer hubs that employ a similar shifting mechanism, such as the TCW and SC3 coaster hubs, the FW four speed and the twin cable five speeds (on the drive side).
Step 1: Select high gear
Select high gear.
Step 2: Take up all the slack in the indicator chain
Back off the locknut. Turn the adjuster until the tiny chain is taut – not tight; just enough to ensure there is no slack.
Step 3: Loosen the chain slightly
Back off the adjuster so the chain is slightly loose, so you can move it from side to side a wee bit without feeling any tension. Do up the locknut.
Explanation: High gear is held engaged by a spring. You want to make sure that spring can do its job, as there's nothing else to hold it in high. Tension in the chain indicates that the spring is being held off slightly and not holding high gear as securely as it should. Slack in the chain tells you the spring is fully released and able to to its job.
Step 4: Check
Select neutral; you should find it roughly midway between second and high gears. With neutral selected you should be able to turn the cranks without turning the wheel. If that is the case, and the shifter is roughly in the middle, then you should be on your way. If necessary, turn the adjuster a smidgen one way or the other.
While you're at it, check that the chain freely slips back into the axle when you select high gear, and that the amount of slack in the chain remains consistent each time. If it isn't consistent, then you most likely have a problem in the cabling.
Note: When changing gear it may be necessary to turn the cranks slightly to allow the relevant parts to mesh.