Zod and Co.... they're just hanging out at the white house.. but society keeps on functioning like normal.
Zod doesn't bother to put a new government into place... he and Non and Ursa just can do whatever they want but have no enforcement mechanism for their rule.
So how did Zod intend to create a New Order?
Thank Lester for the bit in the beginning (Why on earth did Thau leave THAT bit in from Lester???) of the White House scene--- in the Mank script:
INT. WHITE HOUSE OVAL OFFICE - DAY
ZOD stares out the window at the destroyed Washington
Monument. NON stands in front of a wall mirror, wraps
the Presidential flag around himself with a contented
growl. There is a terrible scream from O.S. which
suddenly cuts off. USRA enters through a side door, her
chest considerably more decorated than before.
URSA
I'm bored with this puny planet. The men
here do not even try to resist anymore.
ZOD
Then leave. There are dozens of other
life systems nearby. Find one that suits
you.
There is a knock on the door. They turn.Also, the rest of the Zod speech gives a rough idea: (bits here, not the whole speech)
panic.
ZOD
Today I bring a New Order to your
planet! One which shall last until the
end of time! Each of you...
395 INT./EXT. LINCOLN MEMORIAL - DAY
PEOPLE stand petrified inside and out of the Lincoln
Memorial and reflecting pool beyond as ZOD continues.
ZOD'S VOICE
... each man, each woman, each child –
all will march proudly together in this
New Order! Your lands, your homes, your
possessions, your very lives...
397 EXT. WASHINGTON MONUMENT - ANGLE LOOKING DOWN
CAMERA LOOKS DOWN PAST ZOD.
Police cars with flashing
lights begin to converge in the distance.
ZOD
All of this and more you will gladly
give to me!
ZOD'S VOICE
There is no longer a need for separate
nations in this world, no need for petty
squabbles between one group and
another...
ZOD'S VOICE
All of you will work together, strive,
produce, and sacrifice together - and
all for a common goal!
ZOD'S VOICE
There is now one law, one order, one
ruler who alone will determine your
collective destiny! One force before
which all of you shall kneel forever!
404 EXT. WASHINGTON MONUMENT - DAY
ZOD continues to rant on top of the monument.
ZOD
In return for this submission you will
have my generous protection! In other
words - you will be allowed to live.