I have a Pietta and not an Uberti. Sometimes it may go in better from the right and sometimes it seems in go in better from the left.
And with some guns it may be easier to insert the cylinder before thumbing the hammer all of the way back to 1/2 cock.
Sometimes when it gives me trouble, I will start with the hammer all of the way in the down position, and then I will begin inserting the cylinder simultaneous to gradually starting to thumb the hammer back.
That may be because the hand or pawl is a different length on every gun.
And some guns may be more finicky than others. Sometimes it takes a real slow and delicate touch to learn the travel of the pawl by feel, hammer position and movement.