Well, I think we can be quite confident that in general you don't need a wand to turn into an animal as an Animagus - that's sort of what's so special, and why
'[...] the Ministry keeps tabs on witches and wizards who can become animals; there's a register showing what animal they become, and their markings and things [...] and there have only been seven Animagi this century [...]'
Harry Potter and the Prisoner of Azkaban - p.257 - Bloomsbury - Chapter 18, Moony, Wormtail, Padfoot and Prongs
Indeed, Animagi are explictly described earlier in The Prisoner of Azkaban 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's more, transforming humans into animals isn't, apparently, such a big thing. We see imposter Moody turn Draco Malfoy into a ferret, we see Fred and George's canary creams turn Neville into a canary, we see Hagrid having enough confidence to attempt to turn Dudley into a pig. He fails, but still. We even see people transforming themselves, as when we see Viktor Krum (albeit ineptly) attempt to transform himself into a shark and get most of the way. Hard to see what all the fuss is about the Animagus transformation if these circumstances, unless you can do it at will.
But rather than reasoning about what's probable, we can simply refer to Sirius himself telling us how he could transform into a dog in his cell in Azkaban (also referred to in ʀᴇᴅ_ᴅᴇᴠɪʟ226's answer)
'[...] so when it all became ... too much ... I could transform in my cell ... become a dog.'
Harry Potter and the Prisoner of Azkaban - p.272 - Bloomsbury - Chapter 19, The Servant of Lord Voldemort
But of course, you asked about "this specific scenario" and I have two responses to that. The first is that it's very hard to see why not. Again, the power of the Animagus transformation seems to be that you can transform at will, otherwise the Ministry seems to have about as much need to keep tabs on Animagi as it does to keep tabs on every graduand of Hogwarts, and half of those who flunked.
But more concretely, Sirius explicitly warns Peter:
'But if you transform, Peter,' growled Black, his own wand pointing at Pettigrew, too, 'we will kill you. You agree, Harry?'
Harry Potter and the Prisoner of Azkaban - p.276 - Bloomsbury - Chapter 19, The Servant of Lord Voldemort
Kind of a pointless threat otherwise, really, isn't it? And let's remember, there's nothing that's happened to Peter that we have any reason to imagine would stop him transforming, except that he was forced to transform by the man who's just warned him not to transform again.