The UK isn't a very wild place, at least compared with New Zealand, so there isn't a lot of call for 'public huts', by which I'm assuming you mean shelters which are free for anyone to use in remote areas.
The only place I know that such things exist is in Scotland, where thereI know of are 'bothys''bothies' for hikers. You can find out more at the Mountain Bothys AssociationMountain Bothies Association. They are basic shelters, almost all in remote places, where hikers can use them at will. They are concentrated in Scotland (where most of the UK's wilderness is), but there are some in northern England and the UK. Outside of Scotlandthose areas it's very rare that you can't find some kind of accommodation nearby at the end of your day's hiking.
The nearest equivalent elsewhere in the country is the Youth Hostel Association. The YHA in the UK is a little more geared to hikers than other hostel associations, and maintains a good number of cheap hostels in good hiking areas. Not free but cheap, even by hostel standards. Generally need to be booked in advance, but you can sometimes walk in. (Thanks @MadHatter)