Two separate issues might be at play here.
First, regarding the car: is it that the car can't connect to the iPhone, or that the audio isn't going to the car? You can check this by starting your car and then looking on your iPhone under Settings > Bluetooth to see if the connection has been made.
Things may differ between devices, but I'd offer that occasionally audio goes through my headset rather than my car. In those cases, using the car's controls to pause the audio and then to play again causes the audio to be channeled through the car, rather than the headset. This usually sticks, as well: I can activate Siri with my headset or take a call, and when music resumes it will be through the car (very long calls sometimes screw this up, but the play/pause technique redirects it properly each time).
The second observation that you've noted regarding your Bluetooth headphones (headset?) has to do with a limitation of the headset/headphone devices. The Bluetooth protocol allows eight devices to be connected at the same time, but Bluetooth headsets (and headphones, mice, and keyboards) were designed to only be used with one device. This makes sense: why would you be talking on two phones at once, or listening to music from two devices at once? As such, those devices will only allow themselves to connect to one device at a time. Newer headsets (and headphones that can act as headsets) allow you to connect to two devices at once for call-answering purposes, but redirecting them for things like music would likely require that they have their connection forcefully redirected, as you have been doing by shutting off Bluetooth on one device and redirecting your headphones to connect to another.
So, getting back to your original question: it's unknown at this point, but I would guess that the Apple Watch will have a limit on the Bluetooth connections that it can make and will only connect to one device at a time. It's designed to be an accessory to the iPhone, and a personal device, so I think this is a safe assumption. The iPhone has no issue connecting to multiple devices at once, and should be able to maintain connections to the Apple Watch, your Bluetooth headphones, and your car, all without difficulty. I'd imagine that your car and your headphones are connecting to your iPhone just fine, and the iPhone is just preferentially sending music and/or calls to the headphones - when you get a chance, try what I suggested above. If the connection really isn't going through, then I'd be interested to know (but would maintain that it's a rare case, unfortunately for you!).