Tunein is just an aggregator of radio stations and their corresponding streams. Whether or not a station's stream will work on a specific device, how long it takes to buffer and when it starts to play is beyond the control of Tunein and the channel developer. All three aspects, however, will be influenced by the stream handling capabilities of the device.
To analyse your specific issues, one has to know exactly which stations you were trying to play.
The Roku firmware, specifically the audio playing functionality of the roAudioPlayer and roVideoPlayer components, has a long-standing bug that will often cause live audio streams to buffer for an interminably long time. The bug has been around a long time, has been reported several times in the developer forums, and has been experienced by many developers, including myself, who have verified that there was nothing wrong the audio being streamed. That's on Roku to fix.
A couple of workarounds have been proposed in the developer forums, with varying degrees of success. Some channels appear to have worked around the bug, and stream audio without the long delay. That's on TuneIn to implement.