Since you keep talking about "4G or LTE" I have to ask whether you can get 3G data (UMTS, HSDPA, HSDPA+) or just no data at all?

Was the other SIM you tested with also ATT?

And have you tested your SIM in a different phone?

(BTW LTE and 4G are the same thing. If your phone shows them as different that's ATT programming it to mislead you: I know some US carriers did try to pass off HSDPA+, a 3G technology, as 4G, but that's marketing lies rather than reality).

 

So from what you've described, you're able to send/receive phone calls and texting but it's just mobile data that has stopped working. At least that indicates your cellular account is active and the base cellular connectivity is functional as is, and the SIM card is probably OK since it's fully functional in other phones. The issue is apparently just with mobile data on your recently repaired HTC 10. What kind of problem was this phone having previously and what was fixed on it? That might help determine what to look into.
Also, this may or not even be relevant but if you log into your AT&T account are there any settings/options to add or delete specific phones from your cellular service account? (i.e. My carrier is Ting and there are options to selectively add/remove phones when accessing my online account.)

 

OK, that tends to indicate the SIM card is corrupted some how. Get a new one from AT&T.

 

Since replacing the SIM has already failed I suspect the problem is with your account (or ATT's management of it). So tell them that they've already replaced your SIM, that you've tried it in more than one phone and that other ATT SIMs work fully in those phones (i.e. don't leave them any opportunity to blame the phone, which is the easy way to get rid of you).

I'm assuming that when they swap SIMs there is a delay before it is activated and your number assigned to it: if they can fully activate it there and then then you could stay in the store until it's done and confirm it's ok or show them that that it makes no difference and so the problem is neither handset nor SIM. You might be unlucky and they may have replaced one bad SIM with another, but that seems very unlikely. So I think they will need to do some investigation rather than just swapping things and hoping that the problem goes away.