I've had this exact problem, a few days ago when I was going to invite my wife to share our home setup, but every time I sent an invitation, it just said that the invitation is awaiting confirmation. But the invitation never popped up.
I found a solution, but it will require you to set up your home from scratch again, so be aware of that. Beyond that, there's no guarantee it will help you, but it has helped me.
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Again REMEMBER, everything in your Home will be deleted and you will have to set everything up again!
1. Start by downloading this HomeKit Reset Profile from Apple:
http://appldnld.apple.com/iOSProfiles/HomeKitReset.mobileconfig
2. Open Settings. Just below your Apple ID, you will see Profile Downloaded, tap in and install, and wait a bit.
3. Go back to the main menu in Settings and scroll down and find the new menu item "Home". It is right after Health. It may take some time before it appears. If you can't see it, close the settings app completely and open it again.
4. Select "Reset Configuration" (This will delete your Home configuration)
Wait until it says "You have no homes to reset"
(I followed steps 1 - 4 on my wife's iPhone too, but it's up to you if you want to do the same)
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Now you can open the Home app again. Everything will be gone.
Leave the Home app open without doing anything for a few minutes, then try setting up a single device. It is important that at least one accessory is connected before you invite others to join your home.
After setup, try to invite your wife again, and remember to invite your wife by sending the invitation to her Apple ID email. In my case it worked, my wife's iPhone received the invite immediately and everything has been working ever since.
Finally. If you succeeded, remember to delete the Home Reset profile again, you will find it here:
Settings > General > VPN & Device Management.
Fingers crossed that this works for you too.
