@anth the old breweries at Duddingston / Niddrie are mapped out on the big KML map that I shared previously.
As far as the mill lade goes, just upstream on the Figgate Burn (south) there is a small dam, reservoir and sluice system on the map that feeds the lade.
This lies to the east of where the Woodlands Grove cul-de-sac is now, the course of the burn follows the old reservoir but looks to just be overgrown now.
You can trace the lade running under the road and into a large mill building that is not aligned to either Willowbrae or Duddingston Road. On the 1895 map it is marked as a corn mill. The building is gone and Duddingston Mills street now sits on top of the site. The lade re-entered the burn due west of where Duddingston Primary now is. The mill is marked on 1854 map, giving an indication of age. By the 1908 map the area is still called "mills" and the building and sluice and lade system are still there, but the mill isn't identified, suggesting it may have been out of use. Would be interesting to root about and see if any signs of it remain.