He lost his finger when it was caught in a machine cog.
and Wikipedia says:
One version was that his brother John had surprised him by snapping his jaw shut when they were children, resulting in him biting it off. Another was that it was done deliberately to avoid National Service. A further explanation he gave on his programme Dave Allen at Large was that he often stuck his finger in his whisky glass and it had been eaten away by "strong drink". He also said the cause was repeated brushing down the dust from his suit with his hand causing the finger to be worn away. One of his memorable stand-up jokes was that, when he was a boy, he and his friends would go see a cowboy movie at the local cinema, then come out all ready to play "Cowboys and Indians". Staring down at his truncated finger, he would mutter, "I had a sawn-off shotgun." On his show he told a long, elaborate ghost story, ending with "something evil" attacking Allen in a dark and haunted house. Allen grabbed and bit the attacker, the studio lights came back up, and it was his own left hand.