I'm having the exact same issue. Apple Watch S5 with Cellular. But I'm paying for cellular for no reason, because the watch is incapable of streaming any music or podcasts if I don't have my phone with me. Which completely defeats the purpose of paying for cellular, because the only reason I bought the watch was to go running without my phone. A song or podcast will load for a long time and eventually start to play, but then after a few seconds it starts buffering and won't play for minutes at a time. I've talked to 2 separate senior advisors at Apple and neither have been able to help. I've talked to Verizon, and the problem is not on their end. Apple sent me a replacement watch but that didn't help at all. This is clearly a software issue.
So I've had to resort to downloading podcasts as a workaround, but even that isn't a reliable alternative. The watch is charging with my phone sitting 2 inches away and it says "Waiting..." seemingly forever. Eventually it will start to download one podcast, but after it accomplishes this simple feat, it again says "Waiting..." Waiting for what??? The watch is charging, the phone is touching the watch, and both are connected to WiFi and Bluetooth. What else could it possibly be waiting for?
If I constantly turn the phone on Airplane Mode and turn off WiFi and Bluetooth, it will occasionally wake up and start to download one podcast, but there is no predicting if this will work (or which podcast it will choose to start downloading). And half the time, it doesn't even download the newest podcast episode - it downloads the 4th-oldest available episode and doesn't seem capable of recognizing that a new episode has been released.
I'm jumping through hoops every day to try a workaround for Apple's unacceptable software issues. I keep hoping each new software update will fix the problem, but obviously this is not the case. In fact, despite 9 pages of comments on this discussion page, every Apple employee I've talked to seems baffled to learn of my situation. As if I'm the only Apple customer on Earth who's experienced this. I've been dealing with this since August, and there is no end in sight.
If you're interested in purchasing a $500+ pedometer plus an additional $10/month to send text messages with no streaming capabilities, I can't recommend the Cellular Apple Watch S5 enough.