I am coming from abroad so have no idea how much.
Just to add a figure to benchmark against, last time I got a taxi from Heathrow to home (between Portsmouth and Southampton) in October 17 it was £80, so depending where in the Southampton area you are going somewhere in the £70-100 range would be reasonable.
This was a pre-booked trip, obviously, anything not pre-booked will likely be a lot more expensive.
black cabs are for London only.
Other companies will charge more reasonably.
You might as well get a coach to Southampton or even a train.
The average journey time between Southampton Central and Heathrow Terminals 1,2,3 Bus is 1 hour 45 minutes. The fastest journey time is 1 hour 45 minutes. On an average weekday, there are 24 trains per day travelling from Southampton Central to Heathrow Terminals 1,2,3 Bus.
do your research before you leave. UK's crowded roads are not quicker than trains