In Maleficent, her lover cuts off her wings and she finds it difficult to walk afterwards and uses a stick. Would this be accurate? Or would it be just as traumatic as losing an arm?
It would be as traumatic, in a different way. Angels should be used to both walk balancing the weight of their wings, and occasionally supplementing with a quick wing stroke. Both are now out of the question, so your angel is forced to lean forward to compensate, and use a stick to maintain equilibrium (until he's used to walking wingless).
Occasionally he'll instinctively try to negotiate a higher step than immediately possible, maybe utter a cry of pain when the torn ligaments don't respond, and fall flat on his face.
Since the wings would be connected to various ligaments and muscles in the back, would it cripple him permanently?
This would depend entirely on how the connection is designed. You know that biologically an angel is not possible, because he'd need too large a wingspan to be able to lift, even with hollow bones and larger chest and boobs made of solid muscle: for the same reason, you can choose the anatomical setup you prefer. The whole organization of the scapulae might not be that of a hominin like a man: angels and centaurs are, in all evidence, hexapods (unless you let them have hands at the end of the wings, or the wings themselves be modified arms, like the common bat).
Chances and extent of recovery would also depend on how the wings were torn - if shorn at the base, digging in the back muscles, or farther out; whether cut neatly, or teared away destroying the surrounding tissues; and so on.
In theory there's nothing against wings actually regrowing. Human arms do not grow back, but angels are not men (I remember one short story of such an "angel").