From what I can gather from web searches, the change happened about 59-60, and through that period I'm guessing (British industry being what it was/is) that they used whatever was on hand at the time. Certainly it did not appear to line up with the II to IIa "change", and there is anecdotal evidence of vehicles with different plates on each end during the 59-60 period.

I'd guess that any attempt to pin it down to a month, or even a year, would be doomed to failure! However from my own vehicles I can fairly safely say that there were both military and civilian '59 SWB's that sported Solihull badges.