Nathan Smith took 4-91 as England all out for 354 to give New Zealand an 84-run first innings lead.
Madison Keys has won a third Eastbourne Open title, beating Tatjana Maria 7-5 6-4 in the final.
Naomi Osaka a doubt for Wimbledon after suffering a foot injury in the Bad Homburg Open final.
England 314-6 at lunch on day three of the third Test, still trailing New Zealand by 124 runs.
England through to World Cup last-32 before they take on Panama in New York on Saturday evening.
Belgium beat New Zealand 5-1 to top Group G and progress to the last-32 of the World Cup.
Egypt earned a nervy 1-1 draw with Iran to set up a World Cup last-32 showdown with Australia.