If I remember correctly in PoA during the scenes at the Shrieking Shack, Wormtail was able to escape but to transform into his Animagus form he needed a wand. However, we never see another wizard (McGonagall/Sirius) need to do this.
When is a wand needed for Animagus transformations?
Update due to answers and comments:
In the film he shown to touch his wand to his head. Clip
Willing transformation as an Animagus does not need a wand. Pettigrew needed a wand in his case because a spell was cast on him to transform him into his human form.
I think the reason that Pettigrew needed a wand in this case was because Sirius and Remus forced him to transform into a human:
Black had already retrieved Severus’s wand from the bed. He approached Remus and the struggling rat, and his wet eyes suddenly seemed to be burning in his face.
“Together?” he said quietly.
“I think so,” said Remus, holding Scabbers tightly in one hand and his wand in the other. “On the count of three. One – two – THREE!”
A flash of blue-white light erupted from both wands; for a moment, Scabbers was frozen in mid-air, his small black form twisting madly – Ron yelled – the rat fell and hit the floor. There was another blinding flash of light and then –
It was like watching up a speeded-up film of a growing tree. A head was shooting up from the ground; limbs were sprouting; next moment, a man was standing where Scabbers had been, cringing and wringing his hands.
- Harry Potter and the Prisoner of Azkaban
Since an (unknown) spell was cast on him to force him to return to human form, it can be assumed that he would not be able to transform back into animal form without first lifting that spell.
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.
