The OP had the author right. I am almost sure this is On the Flip Side by Nicholas Fisk.
Lucas thinks his sister is barmy. She spends hours 'talking' to her pets. But when a world catastrophe threatens, Lettice's affinity with animals seems to offer a way of escape...
This review mentions that it's parallel dimension beings being brought over:
A strange, ambiguous tale of communication with animals and parallel dimensions, it contains some pretty harrowing descriptions of society collapsing as animals go mad and inter-dimensional beings invade.
I have not found a reference to a rural community, but this review mentions the protagonists going to a sort of an earlier time, albeit a Victorian one:
Lettice can speak with animals. If she looks into their eyes she can see images that tell her things, and what they tell her is that the world is threatened by Blobs.
Fisk's On the Flip Side is a humorous horror story in which the mad scientist cracks jokes but is a genius, television produces "Rasters", waves which produce the blobs which -- first unthinkingly, then with malevolence--attack animals and humans.
What makes the story interesting--and it took me three goes to get past the first chapter--is that Lettice isn't very nice and is not in the least bit redeemed by the world's eventual realisation that she is telling the truth (she hates it in fact) and that the animals and humans eventually flee Earth for a Victorian analogue where they forget their old world almost totally.
In the penultimate chapter, both "bovos" and "bless the beasts that bless us" appear.
Lucas didn't even bother to give the correct greeting when he entered the bovo shed. He just muttered, BTBTBU' instead of 'Bless the beasts that bless us.' It was dreadful, the way people took bovos for granted.