Question

how will the equation for an object moving with a uniform velocity change ?

Answer 1

For an object moving with uniform velocity, the acceleratio would be equal to 0.
Hence, in laws of motion, you can substitute a = 0 and they will become
Law1 : v = u
Law 2 : s = ut
Law 3: v^2 = u^2 i.e. v = u again.

Answer 2

In uniform motion, the acceleration if zero,
So 1st equation of  kinematics becomes:

v=u+at
⇒v=u
The 2nd equation of kinematics become:

s=ut+1/2 at 2
⇒s=ut