Snakes work well initially but only feed every few weeks.
I watched my cat get a huge rat and then eat it whole - just left the tail. It made me feel a little queasy as it sounded like it was eating a digestive biscuit. That cat has since disappeared but we have two others. One regularly brings us 'presents'. The other is just stupid - literally meows at prey.

Red and Blue Army!

Have just caught another rat in a Fenn trap in the garden but the trap has closed across its shoulder without killing it. Not sure what to do now; leave it to die I suppose, but that seems a bit cruel even for a rat. I can’t get near it (realistically) to finish it off - I’m not sure how I would do that anyway with a metal trap all around it. I lifted the lid of the tunnel and the b’stard went mental, shrieking and spitting at me. Gruesome. I knew I should have bought that air pistol I was considering.

Palace since 19 August 1972. Palace 1 (Tony Taylor) Liverpool 1 (Emlyn Hughes)

foxes will have it.

Kids,tired of being bothered by your pesky parents?
Then leave home, get a job & pay your own bills, while you still know everything.

Some excellent advice on here, we had exactly the same problem: rats were using our garden as a run to get to the neighbour's bird feeder. You've got to eliminate their food source mate so they go elsewhere, it's going to be very hard to kill them all.

Killing them all: traps are great, consider using soap as bait. They (and mice) like it as nest material, not all foods work.
Jack Russell's or terrier dogs are great if you can borrow one (I'm told).
Also, have you considered taking the war to them? If you know the nest is next door in the gamekeeper's gaff, why not you and him lay some poisoned bait down around it's parimeter and go prone with a couple of air rifles?
Forget the suggestion of humane traps, you're just moving the problem on somewhere else. They've got to go, simple as that.