Is this a first- someone answering their own question? I just found their website and it's a good one- talk about a coincidence- not only was their owner called JJ Broughton, he was then joined by JJ Braddock and JJ Bradburn, so the company was known as the JJB's for obvious reasons. Once the business was bought in 1976, the name was kept.
I didn't ask what the 6 JJ's stood for, I had to draw the line somewhere!