No.
The rights to the .uk depend on the situation as it was on 28 October 2013.
If the .co.uk had the rights to the .uk on that date (all it had to do was exist to have the rights) then it will keep the rights forever. They will not pass to the .org.uk or to any other extension.
In other words, so long as it had the rights, it can be deleted and re-regged a thousand times between now and launch day and it will keep the rights.
On the other hands, if the .org.uk had the rights on 28 October 2013, then catching the .co.uk won't give you those rights (though in practice for a .co.uk to be dropping today, it must have been registered on 28 October 2013 so this will never happen)
The only question which Nominet have to clarify is whether a "suspended" .co.uk domain can get the rights to the .uk. In other words, if the .co.uk was registered-but-expired on 28 October 2013, did it acquire the rights to the .uk?
My instinct is "yes" since Nominet's V3 is very pro-.co.uk, but I'd like to see confirmation from them before being 100% sure on that point.
A .org.uk (or other non-.co.uk extension) can NEVER gain the rights to .uk if it didn't already have them on 28 October 2013. It can only lose those rights, or keep them if it had them, but never gain them, no matter what the circumstances surrounding other competing domains.