Southee derailed England with three wickets for 0 runs, putting New Zealand on top.
James Bracey fell for a duck on debut, knocked over by Southee.
Rory Burns and Ollie Robinson share a 63-run stand to rebuild.
Tim Southee picked up his second five-wicket haul at Lord's, breaking the stand between Burns and Robinson.
Rory Burns reached his maiden Test century at Lord's to keep England afloat and eat into the deficit. England were bowled out for 275, conceding a 103-run lead.
Conway and Latham built on the lead till the former chopped on off Robinson.
Ollie Robinson picked up Williamson and Conway as New Zealand ended the day on 62/2, ahead by 165.