Air Carrier Captain reported a runway incursion when their exit from the landing runway was blocked by a taxiing aircraft. Captain reported tail of their aircraft had not cleared the runway when ATC cleared a third aircraft for takeoff.

Narrative

After landing on rwy 32 tower assigned us the Echo taxiway which we complied with. As we tried to get off the runway on taxiway Echo ground cleared Aircraft Y to taxi in front of us preventing us from clearing rwy 32. Our nose was barely across the hold short line and we were immobile while Aircraft Y taxied in front of us preventing from our wings back from being across the hold short line when Aircraft Z was cleared for takeoff. I noticed this after they were about 1500 feet into their takeoff roll and got the rest of the aircraft across the hold short as quickly as possible. After parking the aircraft at [Gate] XX I went to the gate agent to get the towers phone number and spoke to a ground controller about the runway incursion. He knew what I was talking about right away and said they realized after the fact" that there was a conflict but then saw us moving forward so didn't feel they needed to do something. I politely stated that ground should yield traffic to aircraft exiting the runway to prevent a mishap but tower should also be aware of aircraft not clear of the runway. He agreed with everything and the call was terminated.Cause: This is one of two ATC issues at NAS that totally violates standard ICAO practices and I will file a separate report on the other incident. Bad things will happen if this isn't reigned in. Aircraft aren't permitted to takeoff while another is still on the runway. This is how the midair happened- lack of supervision; nonstandard procedures; willful disregard for proper procedures. Unacceptable."