There is no information given in this novel concerning this, so the fate of animals is only a matter of speculation. However, in Lowry's 'Gathering Blue,' another community saves Jonas and Gabe from their plight and welcomes them to live among them.
The existence of a third village is also evoked, the community of Kira, a young handicapped woman who is a skilled weaver. Here each year an appointed Singer ritualizes the story of "the Ruin" in an annual commemorative ceremony which could either be interpreted as reference to either a nuclear holocaust or a global epidemic of some sort. This would account for the sparsely scattered villages not having much to do with one another and the absence of a normal ecosystem, including a diversity of plants and animals.
I would assume that they all got killed. Although we know that there are still fish because the one person becomes a fish hatchery attendant.
I think they got killed because the government decided that they were not useful anymore and that they got in the way of creating the society that the government wanted. The government wanted a society without emotion. But animals can make us feel emotion -- we love our pets, we think forest animals are beautiful, that kind of thing.
Whenever something is not useful to the government, it gets killed. So I assume that's what happened to the animals.