I can find no reference to a North British Militia although the 21st Regiment of Foot, (later the Royal Scots Fusiliers), was known as the 21st (Royal North British Fusiliers) Regiment of Foot, a regular Infantry Regiment.
The 8th Regiment of Foot was recruited in Liverpool and was known as the 8th (Kings Liverpool) Regiment of Foot.
Militia Battalions (3-4 to a Regiment)were raised locally by County Sheriffs for Local Defence, although some saw active service during the American War of Independence.
Militia Battalions ceased to exist after the 1881 Army Reforms and most were absorbed as the 3rd (Depot) Battalion of the Regiments to whom they were affiliated.
However, it is possible the Regiment (or Battalion) you seek was raised under the Command of a local Landowner or Laird.
Can you be more specific about the date he served ?