My gt grandfather married a second time (bigamously of course) during WW2. It was while he was in service. He was put in prison in London for an unspecified amount of time, and we assume it was after he was discharged - I'm waiting on his army records to ge tthe details - it was picked up because his first wife (my gt gran) stopped receiving his wages as he'd presumably provided his new wife's details.
It seems that looking through the local papers, it was pretty rife around the time of WW2. My story was apparently front page news in the News Of The World (don't know why, but hopefully it was a really juicy story!), and other stories I've seen almost always carried a custodial.
It appears that the records of the marriage may have been struck from the registers as I cannot locate it - there was one that possibly matched, but we've rule dit out as not him and just someone with the same name.
If your guy was caught, I would expect there to be a piece in the local paper - our local paper showed no less than 5 in one year, and I live in the relatively unpopulated Welsh Valleys!