I don't know Southampton very well but I see there is a "The Pig..." there. I've been to one elsewhere and it was really good.
In terms of London, I'd say don't limit yourself to the south, you're never really more than 45 mins away from anywhere really. Brick Lane is a fun area, it's kind of hipstery/street food/vintage clothing-y but also has two of my favourite places.
SMOKESTAK - one of the best things to actually survive the barbecue revolution a few years ago. Everything is smoked on premises, no take out.
Beigel Shop. This is one of (if not the) oldest bagel joints in London. It's open 24/7, take out only, they only do three things (lox, chicken & salt beef). Have the salt beef with pickles and mustard, expect to queue, thank me later.
E. Thought Beigel was older, it's only from the 70s. E2. Found it, it was next door. It's from 1855, link updated