Will any phone that worked with Telus or Koodo also work with Public Mobile without having to unlock?
Technically, the answer is no because of an old network and devices that Telus and Koodo used to use, but I'm sure that's not what you're asking.
Also, while the answer is generally "yes" for almost all of the HSPA compatible devices that were sold as Telus and Koodo devices, once again, the answer is unfortunately no if you want an absolute guarantee. I have come across Koodo devices that required an unlock code when inserting a Telus Mobility SIM card. You'll probably never encounter such a device though.
@beedle both models you listed work on PM. Although likely not required to be unlocked it is always recommended to get them unlocked incase you want to use the phone with another carrier in the future that is non Telus. It's quick, easy and free to get them unlocked. Just call the provider it came from.
Assuming the device is Telus-locked or Koodo-locked and that it won't work with Public ... Telus or Koodo must provide free unlock when requested.
Unless the phone hasn't been fully paid off or it's been permanently blacklisted or it's been temporarily blacklisted (reported as lost/stolen/etc). The carrier can then legally refuse to unlock.
Unlocks can also be purchased on ebay, usually very cheaply. Little to lose except the money you spend on them, they'll work or they won't work. Some devices do impose a limit on the number of unlock attempts before they perma-lock or even brick themselves.
If you don't know for sure then just try using the phone with an active PM SIM card. If it works with PM then it works with PM regardless whether or not it's locked.