In my playthrough my character uses Yes Man to create his own government over the Mojave. In it there are several departments set up with different people placed in charge e.g. Department of security headed up by Boone. I have placed companions as heads of different departments I feel best suit them by still have two departments that need to be run by people and have ran out of companions.

The two departments are for welfare which helps oversee the improvement of the lives of citizens and industry which oversees rebuilding of areas and manufacturing of goods. Both departments are closely tied as those who work are granted many welfare provisions and much of the manufacturing is there to improve the lives of Mojave inhabitants.

I need two characters from Fallout who would best be able to run these departments and ideally no one from Legion or NCR as they've been kicked away. Any ideas?

Someone from the Followers and Raul, respectively.

Judah Kreger.

That guy michael angelo could help the guy that oversees the construction thing, to have neons everywhere until you cant see the moon.
For welfare you could give the followers that

Okay, decided to do a reshuffle and place Raul as head of the department of industry. However, now we have no one to run the ministry of justice which controls prisons, courts etc. any ideas for that?

Raul strikes me as more of a follower than a leader. You could give the King some position. He seems genuinely concerned for the downtrodden, and he's already a leader. Plus it would ensure the loyalty of the Kings.

Primm [censored]ing Slimm!

Judah Kreger for security. No other can run it better than a former Enclave officer.

Infact just let Primm Slimm lead.

Vulpes Inculta for intelligence officer.

YES