I can understand why your wife would feel disrespected. I’ve always thought that significant others — especially spouses! — should be invited to weddings (with very few exceptions), but sometimes the definition of “wedding” can be flexible, and this is such a case. Yes, there will be a private ceremony where your brother and his bride will be getting married, but the reception will be much more inclusive and your wife and teenage sons are invited to that. Would it be nice if extended family — which, for your brother means his sister-in-law and nephews — were included in the actual ceremony? Yes, of course. But for whatever reason — budgetary, space, wanting more intimacy — that isn’t happening. And the fact is, you don’t really know what that reason is. Maybe your brother’s fianceée has always wanted to get married in that house where they’ll have their ceremony and there simply isn’t enough room for more than a handful of people. Maybe they’ve decided that this is a super personal moment for them and they want it to be practically private, save for the closest people they grew up with. Maybe they don’t want any kids there.
As hard as it may be to understand your brother’s motives for keeping his wedding ceremony super small, for the benefit of your relationship, you need to let it go. This is his day and since he is sharing it with his larger family and friends in a way he and his fiancée are comfortable with, you — and your wife — need to try to respect that. What you can do is tell him that your wife is disappointed not to be able to see him get married and ask if he has given some thought as to whether and how he might include his reception guests in the actual ceremony, like by sharing some kind of video or photos at the reception. You can even offer to help with that if you want. And maybe by opening up that conversation, your brother will offer an explanation as to why he’s chosen to include “immediate family only” at the ceremony — an explanation that will, hopefully, help smoothe some of your wife’s ruffled feathers. (But if not, please, please try to encourage her to put her pride and hurt feelings aside and celebrate your brother’s nuptials at the wedding reception she has been cordially invited to. It’s unlikely that they’re going to change plans now just to appease your wife, and it would certainly put a damper on family get-togethers in the future if your wife decides to sit out the wedding celebration because she felt slighted).