When instructed by BigTalker, SmartThings converts the text to audio in the cloud and generates a URL to the mp3 audio file, then it sends the url to your musicPlayer device, your musicPlayer device then loads the MP3 from the URL provided and plays it. The delay could lie anywhere along this process.
I am not familiar with the Samsung R6 (nor the SmartThings Device Handler for it). You could turn on Debug Logging within BigTalker, then launch the Live Logging page and see if you can spot the delay either within the BigTalker logs or those for your Samsung R6 device. I would also be curious if there are logs that you can access within the Samsung R6 itself and do those match up with the time it received the URL to play and did it introduce a delay while retrieving the audio file from SmartThings servers?