^^^British Rail???
They've not existed for 18 years!
However, as Jackdaw33 suggests, a taxi (or other local travel arrangements) to Waterloo might be best, where staff will assist the passenger onto the train (using a ramp to access the dedicated wheelchair space). At Southampton the train will be met by staff who will assist with the transfer to the taxi rank there. I've run a railway station and such trouble-free transfers are simply routine.
All London black cabs are wheelchair accessible.
Jackdaw33 has provided the telephone number for SouthWest Trains.
Mobility Taxis provide wheelchair accessible taxis from Southampton Central station: 023 8089 9291