I think you remember incorrectly:
Pettigrew had dived for Lupin's dropped wand. Ron, unsteady on his bandaged leg, fell. There was a bang, a burst of light - and Ron lay motionless on the ground. Another bang - Crookshanks flew into the air and back to the earth in a heap.
'Expelliarmus!' Harry yelled, pointing his own wand at Pettigrew; Lupin's wand flew high into the air and out of sight.
'Stay where you are!' Harry shouted, running forwards.
Too late. Pettigrew had transformed.
Harry Potter and the Prisoner of Azkaban - p.279 - Bloomsbury - Chapter 20, The Dementors' Kiss
So I conclude that Pettigrew did not require a wand to perform the transformation.
The film is the film, but even having watched that clip, I don't see any sign of him performing any magic on himself.
To speak to why he wouldn't have just transformed immediately: In the initial stages after Lupin transformed, Lupin was as dangerous to Wormtail as anybody else - Wormtail was actually tied to Lupin. After Sirius had pulled Lupin away, Wormtail had, amidst all the panic and the adrenaline, to make a decision. Crookshanks, in particular, would have been very dangerous to him post transformation. In that moment, he chose to go for the wand and take out Crookshanks and Ron before transforming when he lost the wand. Presumably, had Harry not cast Expelliarmus against him, he would have taken out everybody else, too, before making his escape.
And so I remain of the opinion that your premise is incorrect. We see elsewhere that Sirius is able to transform into a dog in Azkaban without a wand, we see JK Rowling describe Animagi as:
wizards who could transform at will into animals
Harry Potter and the Prisoner of Azkaban - pp.83-4 - Bloomsbury - Chapter 6, Talons and Tea Leaves
What makes the Animagus transformation special is precisely that you can do it at will. Otherwise it's just human transformation which requires skill, but is not a special ability and does not require registration. Moody is able to transform Malfoy into a ferret, Krum (admittedly rather badly) attempts to turn himself into a shark.
And so I think, really, you're asking us to solve a problem that doesn't actually exist. The answer to your question:
When is a wand needed for Animagus transformations?
Is: It isn't. That's what makes the Animagus transformation interesting.