TJ… if you are quoting Mildred, then get it right… they said:-
The Animals Act 1971 provides that the keeper of an animal is liable for any damage it causes, if he knows it was likely to cause such damage or injury unrestrained.
The bit you seem to have just glossed over is the section that says, ‘if the owner knows it was likely to cause such damage or injury unrestrained’.
In this case I’d say unless oldgit knew the dog was likely to try and escape the garden, and had a habit of killing birds, then quite how he is supposed to know the dog is likely to do that is quite beyond me, so I would imagine he would not be strictly liable under that act.
I would venture to say that it would be a good idea though for oldgit to try and sort something out with his neighbours…