Here's a rough estimate of the data process
Internally in the headband you have:
Electrical pulse on head -> [Muse hardware processing delay] -> Bluetooth Packet (no timestamp).
From the band to the device you have:
Bluetooth packet -> [Device bluetooth stack buffering delay] -> data packet -> [Interaxon SDK processing delay] -> processed data (with timestamp) -> [Muse Monitor processing delay] -> UDP OSC packet
UDP OSC Packet -> [Network Transmission delay] -> UDP OSC Packet received by server.
So it really depends on what you want to measure.
Regarding the electrical signal to Interaxon timestamp creation, the most delay in my experience happens with the bluetooth packet buffering. You can see this just by looking at the timestamp delta between raw eeg samples. There is a clear grouping and buffering going on with one sample taking ages and the rest being very quick.
That aside, I think the best way to measure the whole thing would be to sync your device time with a hardware pulse generator, then send a timed 3v pulse through the AUX port pin 4. You will then have a start time referenced to the device and can measure the delay.
Alternatively, if you have a high speed camera, you could film Muse Monitor's raw eeg view and tap your finger on the AUX USB ground to generate a pulse, then watch the delay in the graph changing. (Enable AUX RAW view in settings)