Invitation to join my HomeKit not getting to my wife.

Thank you for the additional details, Duncan314.

Has she already tried to locate the invitation both in her notifications and the Home app? Let's check both by doing the following:

1. "On the Home tab, tap ..., tap Home Settings, then tap Invite People.
2. Tap + to choose people with an Apple ID from your contacts list, or enter their Apple ID email addresses in the To field.
3. Tap Send Invite.
4. Ask the invitee to do one of the following:

• In the notification: (iOS or iPadOS device) Tap Accept.
• In the Home app: (iOS or iPadOS device) Tap ..., tap Home Settings, then tap their name."

Reference: Allow others to control accessories in your home - Apple Support

If the issue remains, and you're still showing in her Home app, she may want to try removing you and then you can try sending the invitation again.

Cheers!

Hello Duncan314,

Welcome to Apple Support Communities! It sounds like you need help with inviting someone to HomeKit. We completely understand why you'd reach out, and are happy to help.

First, great job making sure the devices are updated, that is a crucial step. Next, are both of you using two-factor authentication on your devices?

Two-factor authentication for Apple ID

If you both already have these things in place on your devices, try restarting your Wi-Fi router, connecting to your network again, and then sending the invite again.

Please have a look and let us know if this helps! 

Cheers! 

Hi Duncan314,

Thanks for letting us know you are trying the update, as that will assure you have the most optimized version for your iPhone. If you still run into the issue after installing that, reach out to Apple Support as this issue may require some closer guidance. You can reach them here: Apple Support

Take care.