The example I gave was the British Labour Party, which I assume is broadly similar to the NZ Labour Party, which is in fact linked with the UK Labour Party through membership of the Progressive Alliance.
There are differences between Labour Parties. As you will know in the United Kingdom, the House of Lords, acts as an ‘upper house’ to Parliament which is considered the ‘lower house’.
An upper house generally has the power to make amendments or reject bills (legislation to be). In the United Kingdom the House of Lords has this power, but generally can only delay bills. The House is made up of un-elected Lords, who are generally appointed by the Queen.
Whereas in New Zealand= doesn’t have anything like that. Once upon a time New Zealand did have an upper house, the Legislative Council. This it was done away with when it was considered un-useful.
This means that bills can pass really quickly in New Zealand’s system.
As you know the United Kingdom , have the traditional first-past-the-post voting system. The way this system works is the country is divided into constituencies. A constituency is a geographical area, with a certain number of voters. Each constituency is represented by a seat in Parliament. To hold that seat, you need to be ‘first past the post’, that is, get the most number of votes in an election.
Whereas in New Zealand we have a proportional representation system, called MMP. In this system, voters get two votes, one for their electorate (equivalent of a constituency), and one for a party vote. What this means at the end of the day is that the make up of Parliament will reflect the overall nationwide party vote. In this voting system it is difficult for anyone party to get 50% of the vote share alone, so coalitions with other parties have to be made.
In terms of forming a Government, both nations are the same - but since the Queen does not often come to NZ - she gets the Governor General to do this for her. The Executive Council does the job of the Privy Council. Again the Queen follows the advice of her PM.
In New Zealand local government elections are not run under the banner of National vs Labour, like the Conservatives v.s. Liberal in the UK. There are left v.s. right factions but the politics is less pronounced.
We do have race based parties here and they do hold representation in Parliament through the Maori seats - the Maori Party. The UK does have parties which represents the various nationalist groups such as the Irish or the Scottish.