My FCOM states:
[737-300/400/500]
Do not engage the autopilot for takeoff below 1000ft AGL
[737-700/800]
Do not engage the autopilot for takeoff below 400 ft AGL.
Since we do mainly fly the classics (first NG arrives in around 10 days) our SOPs still require to handfly on a normal take off or single engine take off until we have cleaned up the aircraft.
Since all ILS approaches are flown with dual autopilots a normal go-around before we switch to manual flight will be fully automatical which reduces the workload considerably. However switching to manual flight is encouraged before 400ft so we do not have to fight the trimming forces. Personally i switch to manual flight way before that (either raw data from the beginning or manual after GS intercept), except if visibility is marginal and a go-around is likely.