You might like to consider this, from Hansard from the House of Lords:
Lord Selsdon asked Her Majesty's Government:
Which mammals and other animals are classified as "vermin".[HL4559]
Lord Whitty: There is no definition of the term "vermin" in UK law. In such a situation the Oxford Dictionary definition should be applied.
The Oxford Dictionary defines "vermin" as "Animals of a noxious or objectionable kind. Originally applied to reptiles, stealthy, or slinky animals, and various wild beasts; now, excluding in US and Australia, almost entirely restricted to those animals or birds which prey upon preserved game . . ."
The Small Ground Vermin Traps Order 1958 and the various Spring Traps Approval Orders, refer to "small ground vermin". Neither the orders nor the Pests Act 1954, under which they are made, define this term or provide an exclusive list of species. However, the following animals are listed under various orders: moles, grey squirrels, rabbits, mink, stoats, weasels, rabbits, rats, and mice.
Traps approved under the Spring Traps Approval Order 1995 do not apply to small ground vermin listed in Schedules 5 and 6 to the Wildlife and Countryside Act 1981. This means that red squirrels, dormice, water voles, shrews, hedgehogs, polecats and a number of other species are excluded.
or this, from a UK Law forum:
There is no legal definition of vermin. Vermin means any species that is destructive or injurious to health.
We are talking about the law here which, sometimes regrettably, overrules any social, cultural, political, religious or other views and beliefs we hold.
Think about how you would feel if a child was bitten by a rat because you failed to deal with the problem, or contracted Weil's disease from coming into contact with rat urine? We don't live in a perfect, ideal world where the species are balanced and haven't done for thousands of years and about the only natural predators of rats in this country are birds of prey and cats. However unlike foxes, they will hardly ever kill wholesale but, as your cat has done, kill one each day.
There aren't many people who enjoy killing things, despite what you read about hunting, fishing and shooting but sometimes it is necessary for the preservation of life on our overcrowded island.
Marigold has researched and found the relevant legislation for you. We're not talking about "indiscriminate killing of wild creatures", this is an eradication of something that could easily become a public health problem.
One last point; have you realised that if it can be shown that it is your poultry keeping which is the source of the problem, you could be barred from keeping poultry at your home?