There is actually a food and drink plan for the next four years from DEFRA and some input from newly formed DIT. This includes an additional £293m of additional sales to Australia and New Zealand. This is without a specific FTA. Up to 2020 these will be supported by EU-Australia and EU-NZ agreements as below
DEFRA can offer support such as access to food and drink trade shows, in-store promotions, a chance to exhibit at trade shows and reduced fees on eCommerce marketplaces.
These exports will be conducted under EU-Australia Partnership Framework of October 2008. This aims, apart from cooperation on the multilateral trade system and trade in services and investment issues, to facilitate trade in industrial products between the EU and Australia by reducing technical barriers, including conformity assessment procedures.
So even before an FTA the UK needs to replace existing EU-Australia/New Zealand agreements to ensure existing trade flows as at present.
As for New Zealand see below. Again even without an FTA, the EU supports UK food and drink exports to New Zealand in many ways.
EU and New Zealand
Since 1999 the EU and New Zealand have a bilateral agreement for mutual recognition that aims to facilitate trade in industrial products between the EU and New Zealand by reducing technical barriers, including assessment procedures. It covers medicine products and devices, telecommunication equipment, low voltage equipment, machinery and pressure equipment.
In 2003 a veterinary agreement entered into force aimed at facilitating trade in live animals and animal products while safeguarding public and animal health, and it has been updated in 2015.
The senior official level Annual Trade Talks meet regularly to exchange views on the bilateral trade relations.
The Leaders of EU and New Zealand, in their meeting on 29 October 2015, announced to start the process for negotiations to a comprehensive free trade agreement.