I'm thinking about having a driver here in Hong Kong , how much do they cost and do you have to arrange a UK work permit for them whilst they are driving for you in the UK?
How much they cost depend on a few factors.
Hired driver - 5000-8000 HKD
If you want a more professional private car driver with experience then it'll be at least 10000+ HKD for a starting salary. Add 2000-4000 HKD if they can speak english. Of course this is just a starting point, many people who have had the same driver for 10-20 years pay there drivers upwards of 20000 HKD.
Not too sure about working in the UK but if it's only for a few days/week, you pay him in HK, and you are on a "business trip" it should be fine.