Broadband dongles from this country would be very expensive to use in another country. Plus I'm not entirely sure that they'll work, since when you buy a 3G dongle, it's usually locked to a certain provider. You would also need to pay roaming charges to use a UK dongle in NZ or an NZ dongle in the UK.
Some of my collegues went to India and they took along with them a USB broadband dongle and a Netbook (a small portable laptop) which they managed to get by with. So it looks like it is possible. The dongle they used was a Vodafone one.
Roaming Internet charges using a dongle can work out to be very expensive so be very careful as you are charged for every Megabyte used and it can be something between £4 - £7 per Mb.
Phone.
My advice would be to buy a mobile phone that is unlocked. I mean that it is not locked to any provider.
The purchase a pay as you go SIM card. You can get them for about £5.00 (GBP).
Use the PAYG sim card for phoning. When she goes to New Zealand she can buy another PAYG sim card.
The other alternative for a phone is to use a Global SIM card which works anywhere:
http://www.globalsimcard.co.uk
Martin Lewis wrote a good guide about using mobiles abroad and you can read it here:
http://www.moneysavingexpert.com/phones/cheap-roaming-calls
Internet.
Using a broadband dongle, you may be able to buy a SIM card for use in the UK and another SIM card for use in NZ which might be cheaper than roaming charges. You'll need to check with the respective providers. Internet Cafe's are also an option for quick, easy and cheap Internet access.
If you have a laptop or netbook with WIFI access, then you can purchase wireless accounts from some providers for use in hotels and such like. There are also alot of free WIFI hotspots in places like MacDonalds, Airports, etc.