Hi Eddie and RWalk, it's Mike. Sorry, I don't have any knowledge of BSA serial numbers or possible date codes.
Bsa Rifle 270 Serial Numbers
From what I can find in reference books, BSA centre-fire rifles came in three versions: Royal, Majestic and Monarch. All shared similar actions, which varied in length depending on calibre.
The 30-06 was built on the longest version of the action, which was introduced in 1957. I don't think any BSA centre-fire rifles were built in Birmingham after 1974, but I am not too sure of that date. I think Bill Harriman might be the man for more detail.