This hits home for me. My mother and I live on different continents and my constant guilt and misery about it is all the more severe at holiday times. My husband is sympathetic, but doesn't really understand my emotional hand-wringing and all the family drama concerning The Holiday Problem.
Your brother-in-law is obviously clueless, as well as rude and totally lacking in empathy. But I'm wondering whether your sister was feeling very guilty that your mother was being left alone and it was this that prompted his horrible comments.
The charitable interpretation would be that he knew how guilty your sister already felt and saw her getting tearful at hearing your mother talking about feeling lonely. So he grabbed the phone to try to stop the comments and thus prevent your mother from upsetting your sister any further. And, being rude and clueless, he irritably said the first stupid thing that came into his head.
The less charitable interpretation is that there had been some disagreement about your mother's being left alone. So he was trying to silence any further protests from your sister about how they would spend Christmas, by bullying your mother into silence.
Does your BIL usually participate in telephone conversations with your mother? If not, it seems to me telling that your sister put your brother on the phone again the following day, even though he'd been so horrible. If my husband had been that rude to my elderly mother. I don't think I'd have let him speak to her the next day.
So what prompted it?
If Theory One is correct, your BIL may well have felt guilty about his rudeness. So he wanted to speak and his question about her day was his clumsy way of trying to make amends. It may seem crazy, but from my long personal experience, it is typical of an "apology" from a socially/emotionally inept man who's not good with words.
If Theory 2 is correct, then I'd be a little worried. Because I'd think that my sister might have been bullied into letting the husband talk. Either over her objections, or at his insistence so that he could crow afterwards that he'd been right; Mother had been fine; etc.
ETA: Your sister may have been telling the truth about his trying to be funny. Although my husband has never been intentionally rude to my mother, he sometimes makes fakes angry comments when I'm on the phone to her, and even though I explain that he's trying to joke about, I'm sure she doesn't believe me. But if I'm honest, I must also confess that occasionally I have pretended to her that a genuinely angry comment from him to me was merely a joke. :-(
[This message edited by Cally60 at 12:56 PM, December 27th (Friday)]