Teams external chat not working after upgrade

Sure, I was focusing on this part

If I press the link on "here", it opens a new teams chat window, this time with the skype for business logo

I would do the following to make sure it works

1.) Ensure, through Powershell that both of your users are 100% Teams only on the back end

https://docs.microsoft.com/en-us/powershell/module/skype/grant-csteamsupgradepolicy?view=skype-ps

You would use something like this

PS C:\> Grant-CsTeamsUpgradePolicy -PolicyName UpgradeToTeams -Identity mike@contoso.com

2.) Check it and make sure you are both on your respective lists in your tenant

Get-CSOnlineUser | select UserPrincipalName, teamsupgrade*

3.) Start a completely new chat with the other person

That 'ought' to fix it, may take a few hours to propagation and you may have to check back a few times in a 24 hour period, but from my experience of this issue, it should help to fix it. If it doesn't you can go down the routes of clearing the cache or re-installing the client

It is well known doing the upgrade in the TAC doesn't always have the desired effect and so falling back on the shell to re-propagate fixes a lot of it

Let me know how you get on

Best, Chris