Try uninstalling the Iplayer app then installing the bbc media player app first. Then reinstall the Iplayer app, it does work with both sd and hd versions of the fire. There also is a known issue with some wifi connections which the BBC have said will be fixed in the next update.
BBC only blocks the IP addresses of some VPNs. Good VPNs still work with BBC iPlayer. Follow the steps below to and you’ll be able to watch BBC iPlayer (again) from anywhere in the world. Upgrade your VPN to ExpressVPN ($6.66/month) or Ivacy ($1.16/month) or another VPN that still works with BBC iPlayer (see below) Download and install the app
2011-8-1 · I can't get BBC iPlayer on my HTC Legend. I get a message that says I need to download Adobe Flash Player from the Android Market but when I try to do so it says ‘Not Found’. Why can't I find BBC iPlayer in the Google Play Store?
Easy Fix: Can't Download Geo Locked Apps On iPhone/iPad
2020-7-20 · On BBC iPlayer, tap the Menu option at the top-left and then, under Channels, select One. (As far as we know - and barring sudden breaking news stories such as a prime minister resigning BBC - iPlayer Radio