The relevant legislation here is the Dangerous Dogs Act 1991, particularly Sections 3 and 10.
If a dog is dangerously out of control in any place in England or Wales (whether or not a public place)—
(a)the owner; and
(b)if different, the person for the time being in charge of the dog,
is guilty of an offence
For the purposes of this Act a dog shall be regarded as dangerously out of control on any occasion on which there are grounds for reasonable apprehension that it will injure any person or assistance dog, whether or not it actually does so...
Notably, this act only covers your dog being a danger to humans and service dogs. For it to be deemed dangerously out of control and you to receive criminal charges, there would have to be a reasonable concern that it could potentially attack, for example, someone's child.
I can't find any precedent where an owner being unable to prevent a dog attacking a smaller animal was used as evidence of it being a danger to humans. You'd probably avoid criminal charges.
This one's much more straightforward. Allowing, through negligence, your neighbour's cat to be attacked andinjured by your dog could leave you liable for damages. The most likely damages would be the related vet bills.