If you want a more authentic fish and chip experience in central London, try one of these 2:
http://poppiesfishandchips.co.uk/
http://www.goldenunion.co.uk/
Otherwise find a pub and most should have it on their menus. The trouble with a lot of pubs is that they are part of chains and most things are brought in chilled/frozen and assembled/heated on site. Most of the time it's fine, just nothing special.
As above, you really don't have a lot of time and a lot will depend on when you arrive in central london and where you stay. I'd just do a walk from say Oxford Circus, down Regent Street to Piccadilly Circus, across to Leicester Sq and then onto Covent Garden and then onto the Strand and finally Trafalgar Square. That should be 1 hour at a fast pace or 2 hours for a more leisurely route. You'll see the key sights and have an opportunity for a bite on the way.