The TraceTogether App uses Bluetooth to perform handshakes with other TraceTogether devices. Your Bluetooth-enabled phone is capable of connecting to multiple Bluetooth devices simultaneously, e.g. smartwatch and wireless headphones. The different connections are separate and should not be affected by or affect TraceTogether devices.
If you’re experiencing interference when using your Bluetooth device, you can use the “Pause TraceTogether” feature on the home screen (currently only for Android).