Bicycle delivery drivers certainly seem to favour the illegal bikes that also don't require pedalling (which is one way to tell). Sadly these people riding on the clock are a nightmare!
My bike is limited to 15.5mph and that's fine for my needs.
I'm aware of a type of electric "bicycle" used for cargo deliveries which has no physical connection between the pedals and the wheels. The pedals are on a chain to a freely-rotating hub with a rotation sensor, which then sends a signal to the electrical system to apply propulsion. The signal is binary i.e. as long as you're pedalling (regardless of pedal speed) the system will try to reach maximum speed (~15mph, see below). The hub is independent from the rest of the system and as such is never under any resistance beyond that of the rotation sensor.
The issue with this setup is that when you start pedalling there is a delay while the system starts up, where you are pedalling but not getting any forward propulsion. To combat this there is a separate thumb-operated throttle button which provides immediate forward propulsion, for use e.g. when pulling out at junctions.
The thumb-operated button will only propel the bike up to about 3mph while the speed using the pedals is capped at about 15mph. There is no speedometer fitted to the bike which you are able to see.
From this setup it feels to me like a low-speed electric motorbike but they've just added in the pedals to make it comply with the bicycle legislation.
The bike has "wing" mirrors although these are tiny and are mounted to the handlebars so are useless if you apply any steering at all.
The bike has directional indicator lights although these are mounted only on the rear, therefore oncoming traffic is unable to see you indicating unless you use hand signals.
So yes anyway, to answer the question "how fast can electric bikes go", the one I've mentioned above is capped at, at most, 15.5mph, and due to the pedals being mechanically separate from the wheels I would imagine it would be impossible in practice to exceed this speed.