Another question, when Richard III's son Edward assumes the throne, would he be Edward VI, or Edward V?
I think it would have to be Edward V, if he is called Edward VI that's an admission that Edward V was a legitimate king and Richard III was a usurper.
That said, I think Perkin Warbeck called himself "Richard IV" which would seem to recognise Richard III's kingship. Though you can probably put that down to the fact that he drew support from some of the people who recognised Richard III (like Margaret Duchess of Burgundy).
And Lambert Simnel, in his guise as the Earl of Warwick, was declared "Edward VI" and his supporters included Ricardian loyalists like Lovell and Lincoln (and Margaret). So, you could read that as a tacit admission that Edward V's reign "counted".
However, both Simnel and Warbeck were focused on deposing Henry Tudor and not on the legal minutiae regarding the events of 1483 and what their regnal number should be. Our Edward's claim to the throne is far more closely linked to Richard's than Warbeck or Simnel's, and he probably won't want to do anything that could be seen to jeopardise his father's royal title (especially given his uncertain position).
Also, whether or not
Titulus Regius remains on the books might play a part in determining if Edward V's reign is counted.
Or I'm reading far too much into regnal numbering, and it doesn't matter.
So if Edward married his cousin Elizabeth, would this take place just after his father's ascension to the throne or partway through should things look as if they're going roughshod?
I think, given some of the issues with marrying Elizabeth, Richard wouldn't rush into organising a marriage.
He'd probably try and have his kingship stand on its own two feet, then when he feels that this isn't working or people aren't accepting him he'd resort to the Edward/Elizabeth marriage.
There's also the practical aspect- he'd have to coax Elizabeth Woodville and her daughters out of sanctuary, which took several months IOTL.
He'd also want to wait a while and see if there's any better marriage options on the continent, and in any case Edward is only ~13 when his father ascends the throne which is maybe a little young.