According to the OSMwiki access=no (https://wiki.openstreetmap.org/wiki/Key:access#) tag prohibits general access to the road. “Stronger interdiction than access=private. Examples: a fully closed road; a restricted military facility.” In my opinion, by the level of access to this correctional facility we could equate it with a military base. So, yes, routing will ignore these roads. The only way to indicate that the road isn’t restricted for certain vehicles with the permission is to add the motor_vehicle=private tag, but for the navigation this tag is superior to access=no and the road becomes drivable for everyone. So it will be highlighted on the map but routing won’t be accurate. 
So I can see 2 potential approaches to tag the roads. We may add the motor_vehicle=private tag, but it opens the roads for general access or the second one is to leave everything as it is now, but navigation wouldn’t work on this territory for all drivers. I find the option to leave the roads with access=no more preferable, so nobody could be incorrectly routed to the area. However, we can go with either of them.

I will be happy to know your thoughts and figure out the best approach together! 