Jonathon Green, The Dictionary of Contemporary Slang (1984), has this brief entry for batchy:
batchy a. silly, stupid [Alan] Sillitoe[, The Loneliness of the Long Distance Runner (1959)].
Paul Beale, Partridge's Concise Dictionary of Slang and Unconventional English (1989), has this:
batchy. Silly, mad: Army, C.19–20; RN, since ca. 1910; thence to th RAF, who gave it to the (then) Fg. Off. R.L.R. 'Batchy' Atcherley: as a member of the Schneider Trophy team, he broke the existing air-speed record in 1929.
So if your grandmother grew up in a Navy or Air Force family, she might have picked up the term there. Batchy also appears in Thomas Lyell, Slang, Phrase, and Idiom in Colloquial English and Their Use (1931):
Batchy : (W) mad, silly, (F) "Leave him alone and pay no attention. He's quite batchy, but perfectly harmless."
Alan Munslow & Rober Rosenstone Experiments in Rethinking History (2004) compares batchy to other "mad"-related terms as follows:
The following adjectives mean "mad" in some degree or other. Batchy is mad, or merely silly; etymology extremely doubtful. Batty almost certainly (batchy just possibly) comes from bats in the belfry, and the rhyming scatty is related to the Derbyshire scattle (easily frightened) and the obsolete Yorkshire scatterling (a heedless person); both batty and scatty mean quite mad. ...
One of the earliest related instances of batchy in a Google Books search is from Frank Bullen, The Cruise of the "Cachalot": Round the World After Sperm Whales (1899):
Mr. Cruce, the second mate, had got a whale and was doing his best to kill it; but he was severely handicapped by his crew, or rather had been, for two of them were now incapable of either good or harm. They had gone quite "batchy" with fright, requiring a not too gentle application of the tiller to their heads in order to keep them quiet. The remedy, if rough, was effectual, for "the subsequent proceedings interested them no more."
Alan Bennett uses the term in his memoir, Telling Tales (2000) [snippet]:
the cause of arthritis and has written telling them to abandon their research and just cut off the feet of their socks. Now he daily expects a reply in which they make over to him their entire endowment.
'He's batchy,' says Dad, meaning he's crazy but, as a child, I don't think Uncle Norris's ideas are particularly mad or even eccentric;