Based on the script here (Star Wars: The Empire Strikes Back) there is no evidence of any long periods of time between his hand being chopped off and a new one being attached.
The Falcon soars into infinity and away from the huge Star Destroyer
which seems, by contrast, to stand still.
This is just after picking up Luke, then a few lines down they dock on a huge Rebel Cruiser.
EXT. SPACE - REBEL CRUISER
The Millennium Falcon is attached to a huge Rebel cruiser by a docking
Then the next we hear from Luke (a few more lines) he is being worked on by a medical droid.
INT. STAR CRUISER - MEDICAL CENTER
Luke speaking into the comlink as a medical droid works on his hand.
Leia stands near him while Threepio and Artoo look out the window.
Now as for the functionality of the hand I think this is also evidenced in the script after it is repaired he seems to be able to use it fully. (Of course it may take a bit of lightsaber training for full previous prowess)
Luke looks down at his hand. A metalized type of bandage has been
wrapped around his wrist. The medical droid makes some adjustments in
a tiny electronic unit, then pricks each one of Luke's fingers.
Luke wriggles his fingers, makes a fist, and relaxes it. His hand is
There's not much more to say, It seems to say that it was completely functional after it had been integrated with him. This seemed to happen in minutes to hours instead of days.
At the start of the next movie the script describes Luke as having returned to Tatooine without any previous mention of his name in the script it is safe to presume that he did not do any intermittent training of his hand.
Luke Skywalker has returned to his home planet of Tatooine in an
attempt to rescue his friend Han Solo
Star Wars: Return of the Jedi