Thameslink to KC-T or Midland Mainline to St P. The tube access is closed in KC-T so you have to walk up Pentonville Road to KC proper. I would get the tube to Waterloo and get a train to Southampton Central from there (if you go Thameslink get off at Blackfriars and get the circle line to Embankment then on to Waterloo). You can go Paddington then Reading then Southampton. You can also go to Southampton from Clapham Junction but these are slow.
Thank you very much for your answer j2buttonsw. I didn't end up going to Southampton the other night, but am hoping to go in the next few weeks, so will make a note of your answer for future reference! Cheers.