Since it's FR you're flying and if you're connecting to NZ on a restricted ticket, I'd have to advise flying into London the day before. 4'15" isn't that much time to get from STN to LHR and I'd say it'll be tight even if your FR flight arrives on time.

While FR is fairly reliable, delays are not unknown and they will not reaccomodate or do anything for you in the event of a severe flight delay/cancellation ("you look after youself in return for low fares" is their customer service policy). Since they have tight turnaround times, any delays will cause delays for that aircraft's flight all through the rest of the day.

Fastest way, if the M-11 and M-25 are not clogged up is by bus but these are not all that frequent.

Other ways would be to take Stansted Express to Tottenham Hale (25 mins), change to the Victoria line 22 mins) there and change at Kings Cross for a Piccadilly Line train to LHR (40-55 mins). Train time would be nearly 2 hrs, not including waiting.

Not much faster and more expensive would be Stansted Express all the way to Liverpool Street (45 mins), Circle or Hammersmith & City line to Paddington (20-25 mins) and then Heathrow Express to LHR (15 mins).

See www.baa.co.uk for your options.