In East Lothian, old shed wood is not eligible for collection through the bulky waste service. The council's bulky waste collection does not include building materials, which encompass items like shed wood, so you'll need to find an alternative way to dispose of it.
If you want to dispose of garden waste like twigs and small branches, those can be placed in the brown bin for garden waste collection. For a more comprehensive disposal solution, consider visiting your local recycling centre.