The BBC did indeed build the functionality into a recent update
We recently heard from some folks who'd been digging through the BBC iPlayer app that Chromecast support had been built in ahead of a launch in the UK. Turns out that was indeed correct, and as of today the app is 100% ready to cast. After opening it up you may find you need to wait a minute, or even relaunch the app – as I had to – in order to kickstart things.
But once things are set you'll see the standard Chromecast icon across the top of the apps main interface. Getting things from device to TV are simple, though you need to connect the app to the TV first. Tap on the Chromecast icon, select which room you want to pair to, and you're done.
From there, just go ahead and select what you want to watch and let everything else work its magic. I've been playing around with it for a little while now and am really happy with it. It's pretty quick to start playing, the video quality is good and you can even mute the video from the notification tray on your Android device. Pretty slick.
All you need is the current version of the BBC iPlayer app which you'll find at the Play Store link up top.