There are times in which ticket coupons (especially those downstream- ie returning flights when the outbound has already been used) are under airport control- especially hours before departure- and some times (though rare) TAs may not be able to do anything with them. What the TA should've done at least, was get into your PNR from their end of the GDS, cancel you out of the return segment on Friday evening (thus avoiding a no-show), confirm you on the Saturday evening flight, and then either sent you to BA customer service (phone or in person if you're already there) to either have them re-ticket directly or release the ticket for the TA to reprice and reissue to match the Saturday evening departure.

Honestly, with all the commoditization of TAs over the past few decades and non-existent margins thus TAs now needing to charge caller service fees, this is one of the times where TAs really come in handy. IMO just looking at your story, the TA is likely at fault. The very fact that you were called a no-show for Friday evening tells me they didn't even touch your record to cancel out Friday night's flight (or they did and didn't end the record with the new flights/just ignored their changes once they saw they couldn't reissue).