I have been having the same issue as well, and have been trying to isolate the issue with an Apple support case office. The Apple support case officer has been working with one of their engineering teams in Singapore to assist as well. Apple have been really helpful in trying to isolate this so I thought I'd go through the issues that I reported, and what we've been trying to do to mitigate them.
TLDR; We haven't solved it yet, but we're on the phone every day or two to try and capture crash logs that are causing the issue, and trying new recommendations from the software engineering team. As for BMW (Australia), they haven't been able to help or be willing to try and escalate the issue to their software dev team for further investigation.
These are the issues I've had with an iPhone 11 Pro, iOS 13.3, and a Mini:
CarPlay not starting up when the car is turned on
This seems to be recreated most reliably when the car is connected to another known network (e.g. My home network or work wifi network).
Though, this weekend, there was once instance where I saw it when there weren't any known Wifi networks around. When stationary in the carpark, I've gone into settings to see that the phone's Wifi settings show that it has connected to the car's Wifi network (Mini), but I need to navigate through the car's entertainment unit to manually force a connection to the device to start CarPlay.
Music stuttering issue
This has been well documented in the thread. I've tested this with music and podcasts that were stored locally as well as ones that had to be streamed and haven't noticed any difference.
I only have seen this while driving, but when the music stutters, there is a "spinning wheel" in the top right hand of the music screen. This is why I thought it might only be an issue with music that is being streamed directly from the internet.
One thing I have noticed though is that this issue happens far worse while I'm driving in areas with a "known" network (e.g. Driving past a university with eduroam makes the music stream far worse than just the occasional stutter.
Phone call causing the entire entertainment system
I've noticed that when I've been on a phone call and left a Wifi network (e.g. leaving work), CarPlay crashes (laggy UI followed by a black screen), and then the entire entertainment system crashes. The entertainment system reboots with the Mini logo, and then CarPlay restarts.
What we have tried...
- Forgetting the car on the iPhone, and deleting the phone from the car
- Renaming the iPhone 11 Pro
- Forgetting the two main Wifi networks off my phone
- Testing with music/podcasts stored locally and streamed off the internet (no difference)
- Turning off "Ask to Join Networks" and "Auto Join Hotspot" (Settings >Wifi)
- Turning off "Screen time" (Settings > Screen time)
- Collecting syslog files where I've recreated the crash multiple times for the team to analyse
- Ensuring the car has the latest entertainment system firmware
- Resetting the car's entertainment system firmware
What about BMW?
I have suspected that this is mainly an iOS/iPhone hardware issue, but also suspect that there are issues with BMW's OS firmware that need to be resolved (e.g. additional error handling causing the entire car OS to crash). I've tried reporting the issue to BMW with the hope that this could be potentially reported, and hoping that I could send crash logs to the development team..
The BMW dealerships referred me to BMW head office Australia, head office referred me to the dealerships, the dealership referred me to head office, and now I've been referred back to the dealerships. The dealerships have verbally said, "There is an issue, but BMW Australia head office have not acknowledged it so we can't do anything". They have offered to install the latest firmware in the car, but we've confirmed that this has already been done.