I too am experiencing major problems with the BBC iPlayer Radio app. After the first week impressions are not good at all. The problems I am having are: (1) Stuttering performance of catch up radio on 4G and fast wifi connection. (Yesterday with all the stuttering / stopping and starting it took over an hour to listen to a 45 minute programme.) This is not a one off and has happened every time I have used the app. (2) Battery drain both whilst listening and for some time after listening. I have found the only way to stop this is to X out of the app, switch on airplane mode for about ten minutes to kill data connection and wifi and then start again. I have so far been unable to use back arrow to close the app after listening. (3) Crashes at least once in every session I have used this and will not resume from place it had crashed. (4) No way to see / save favourites.
Interestingly I have been reading reviews for this app in the Apple store and many who have commented there mention similar problems, so perhaps apple is no longer the BBC's favoured platform after all. The android app seems fine and gets excellent reviews. I have a Kindle and the app on there works perfectly and whilst at home I will now use my Kindle or laptop for catch up radio. I have also looked at alternatives. Radioplayer is fine for live radio and podcasts but very limited catch up content. Tune in Radio again fine for live content and offers better performance for live radio (Test Match Special!!!) than iPlayer App. Listen again is very stable and offers all catch up services (eventually) but via Bluetooth or without headphones it is barely audible (as one commentator said it sounds like it is "at the bottom of a well") which is a shame as otherwise is the best alternative service.
Sorry to rant, but as live and catch up radio is one of my main uses for my phone this is a real problem to me. If it is not sorted soon may start looking at alternative platforms. I am going to try to contact BBC directly.