War Domain Cleric, three attacks per turn, does this work?

Attack with a Shortsword, War Priest allows me to make a bonus action attack, have the second weapon be a Light Hammer or Scimitar, Nick property allows that attack to be made as part of the attack action (at advantage because of Vex from the shortsword attack). Use the now still free bonus action to cast Spiritual Weapon for the third attack. This seems right to me but I get mixing rules wrong sometimes, what say you?

Sort of. I think you have misunderstood one thing, but it doesn't stop this from working.

The Nick weapon mastery applies to the bonus action attack that you get from attacking with a Light weapon. The bonus action attack granted by War Priest is a different thing, and Nick has no effect on it. However, you can still do this sequence:

1. Take the Attack action and attack with your shortsword.
2. Make your Light property offhand attack with a Nick weapon, also as part of the Attack action.
3. Use your bonus action to make an attack with either weapon via War Priest, or cast Spiritual Weapon and make its attack.

It should be noted that all of this is contingent upon you actually having access to the Nick weapon mastery property. Clerics (even War Domain Clerics) do not have a feature that grants weapon masteries, so you would need to get it from some other source, such as multiclassing with a class that grants it, or taking the Weapon Master feat.

You're correct. I forgot about needing to having Masery, I keep thinking Mastery and Proficency are the same. I checked the character sheet I'm trying to build this character on and the Mastery Icon is not lit for the shortsword. I know multiclassing would allow me to do it but that is what i was trying to work around. Thanks!