The Rifle, Short, Magazine Lee-Enfield Mark III and Mark III* were renamed in 1926 as Rifle Number 1 Mark III and Rifle Number 1 Mark III*.
As originally manufactured, they were from:
RSAF Enfield, Enfield Lock, Mdx., England, whole period of mfg
BSA (Birmingham Small Arms) Small Heath, Birmingham, England, whole period
LSA (London Small Arms) London, England, whole period
Lithgow (made in Australia, mates), whole period
Ishapore (made in India, sahib), whole period
Standard Small Arms, Birmingham, England, 1915-1917
National Rifle Factory, Birmingham, England, 1917-1918
That makes 6 factories, 7 manufacturing stamps. SSA and NRF were the same factory, before and after nationalisation. SSA and NRF rifles were marked on the rear of the receoiver of the rifle; all others on the wrist of the action (the butt socket).
You have NO idea how long it can take to get all seven!!!!!
Have fun.