Thanks all for your help with this. In the end it was actually a network issue which was really hard to debug, and so stackoverflow friends didn't have all the info - sorry!

Th actual solution: All of the computers run the the same VPN, but it turns out MSS clamping was not enabled on some routers, which meant the VPN dropped packets. Strangely, all other network activities worked fine, it just caused an issue with the Azure TTS API - weird!

(MSS Clamping is a network setting that adjusts the Maximum Segment Size (MSS) of TCP packets to ensure they fit within the limits of your network path, especially when using tunnels like VPNs.)