The tax due on the company car will depend on the taxable benefit deemed to arise from its being made available to you for private use. It is a bit like the salary/car allowance you currently receive. If the benefit were only (say) £3,000pa then it might not exceed the HR Threshold and it would all be taxable at only 20%. If the taxable benefit proves to be (say) £6,000, then more of that benefit will be taxable at 40% than on the car allowance you pay now.
If and when you get a company car, depending on how the company deals with notifying HMRC, it may take a while for that to filter through to your monthly "pay packet". The company may start to make a deduction automatically, or it may instead notify HMRC who will in turn change your tax code to deduct more tax every month.
Company car benefits can be "expensive": the benefit is based on the list price of the vehicle when new, so even if you got a 10-year-old banger worth only £5,000, you would still be taxed as if it were a brand new vehicle. Also, it is generally based on the vehicle's CO2 emissions and the percentage applied creeps up a little every couple of years or so. Finally, the taxable benefit applies whether you do 10 private miles a year, or 10,000 private miles. While your daily commute generally "counts" as private mileage, it can feel very unfair. Having said that, you don't suffer Employees' NICs on the company car benefit - unlike with simple salary / car allowance.
I trust this is useful.