Learn English – BrEng expressions to describe a man who is becoming stupid

For an emphasis on the progression of cognitive impairment, there are lots of "going" and "losing" phrases.

The most animal-related one I can think of that is used e.g. for the increasingly-senile elderly is to go batty. Dictionaries seem split on this: though the key element they all agree on is an aspect of mental instability or madness, several also include the notion that a batty person is eccentric or unpleasant, which seems important in the senses I've seen it used ("batty old cat lady", for instance). According to the Oxford Dictionaries, the "bat" in "batty" doesn't actually refer to the flying noctural mammal, but to a second sense of "bat", an unattractive or unpleasant woman, that may be derived from French bat (prostitute) or perhaps from "battleaxe". Note that "batty" has a completely different meaning in Jamaican slang, related to backsides and hence to homosexuality.

Go dolally or indeed doolally, doo-lally or doolali is from British Army slang for the Deolali sanatorium, Marashtra, India. Originally an insane person was said to have the "dolally tap" (with various anglicisations of the place-name) but while I have often heard "go dolally" I have never heard the original "tap" form, which is certainly archaic. (The linked source says that even "go dolally" is archaic, but I have heard many older people use the phrase, and younger people seem to understand it.)

There is also go round the bend for loss of sanity (from exasperation, intoxication or decrepitude) or, with more emphasis on eccentricity than insanity, go round the twist. To go loopy can capture the crazy eccentricity of "round the twist", or a sense of confusion and befuddlement.

Go off one's head/go out of one's head indicates that someone has become mad or demented (and "off one's head" is often used for states of intoxication, e.g. "off his head on LSD"). There are related terms go off one's rocker or off one's trolley or off one's nut which, in my experience and from a few internet searches, are much more rarely associated with intoxication and more often to cognitive impairment from age.

We also have losing it and variants such as losing the plot, losing one's mind, losing one's marbles, losing one's senses. Note that "losing it" is very general, and often means losing one's temper or emotional control. But it can also refer to a reduction in aptitude or skill, which is useful if you want to emphasise the loss of practical intelligence: "Tiger Woods has lost it", for example.