I may have read this incorrectly.
As I understand the rules, if the whole booking is on one ticket and the published minimum connection time has been met, whilst it may be inconvenient to miss the connection, EI as the delivering airline are responsible for any enforced stay in LHR.
You may be able to persuade EI/NZ that it is in their interests to change the tickets, for free, to avoid possible costs for a 24 stay in LHR.
My two penneth (eurocenteth doesn't sound so good!!).