Your potential problems will be airline ones. "Theoretically" you require:
Arrival in Singapore - proof of onward travel out, by air, within 30 days of arrival under the visa waiver scheme for a UK national.
Arrival in Malaysia - proof of onward travel out, by air, within 90 days of arrival under the visa waiver scheme for a UK national.
Arrival in Thailand - proof of onward travel out, by air, within 15 days of arrival under the visa waiver scheme for a UK national if arriving by land or 30 days by air.
Your only problem in reality will be wether your airline from NZ will allow you to board on a one way ticket to Singapore without proof of onward travel by air. Check with them soon
If you get to Singapore and are travelling by land to Thailand you will not have a problem. The rules are not enforced by immigration but by the airlines. If any legs are by air you may have the same problem.
