Will the rooting method I previously used with my Nvidia Shield Tablet work with the new update? I'll soon be getting a replacement for my Shield Tablet due to the recall.

I used a rooting method that used ADB and Fastboot to unlock the bootloader, flash CWM, then flash SuperSU and root.

It worked with Android 5.0.1, shield update 2.2, but the new tablet will most likely be on update 3.1, Android 5.1.

So I ask - will the rooting method and all of the files linked to in the guide (which I will link to down below) work with the new setup?

Guide I used: http://www.ibtimes.co.uk/how-root-nvidia-shield-tablet-android-5-0-lollipop-install-cwm-recovery-1481041
Your original rooting method will work. The bootloader is still unlockable on the replacement devices, just as it was on the original ones. The fact that it is running a newer version of Android doesn't really have any meaningful bearing on this since SuperSU is still actively developed to ensure compatibility with newer Android builds.