Each of the big factories would have had its own Home Guard unit. See in particular the various pages about Kynoch, a major munitions manufacturer at Witton, via the links below. These units were responsible for defending their factory against attack by German forces - paratroops or land-based. Smaller factories without their own HG units would all have fallen within the area of responsibility of one local HG unit or another. I'm not aware of any MP involvement in these civilian factories.
On a day-to-day basis, for more general security, I imagine that factories relied on more traditional methods such as their own works police. I can remember the Kynoch works police outside the gates at Witton, all fully armed. (Scared the death out of me as a young child!)
Chris