I was talking to a cab driver in Bristol about that and it is apparently because drivers were getting robbed by people strangling them with the seatbelt.
I had the same conversation with a private hire in Salford. He said that it's easy for someone to strangle you from behind with the seatbelt.
My dad never brought that curious aspect up (ex-taxi driver of ten years) but you can pick up dangerous people. I remember when I was younger, he had to take me to the police station as he identified someone in a photo book. He said at the time it was a 'runner' but I later found out it was because of a man who he had dropped off who went on to attack his ex-wife and ex-mother-in-law with an axe, which he had the whole time in my dad's car.
Knowing about these sorts of experiences, I can understand why they would forgo protection of a seatbelt when they potentially pick up these types of people.
It wasn't all bad though. He has fond memories of picking up famous Manchester Utd footballers, his favourite team, before they were famous, e.g David Beckham, Ryan Giggs, etc.
In Australia this used to be the case (same justification of taxi driver robberies), but I believe it's now only legal in Queensland - the other jurisdictions have all changed their rules in the last few years. The taxi driver lobby groups have not liked this.
Also in Germany, where the driver is allowed not to wear a seatbelt while carrying passengers, but has to wear a seatbelt as long as the taxi is empty (passengers are always required to wear seatbelts).
http://www.driving-law.co.uk/offences/seat_belts-exemptions....