If you book a mini cab, they will quote you a price for the journey and it will not matter which way you go as long as they understand what time you need to arrive at the airport by.
You will have to travel along the A23 instead of the motorway, either northbound or southbound depending on where you are travelling from
There will be an official diversion in place which your cabbie should follow, as it will be well signposted you will know if he is going the right way
You have not mentioned where you are coming from,it would be handy to know.