None, for two reasons:
Elizabeth Windsor is descended from Elizabeth of York, daughter of Edward IV -- Richard III would be her so-many-times-removed-I-don't-want-to-count-it great-uncle.
The Queen is a member of the house of Windsor, which succeeded the house of Hanover when Queen Victoria married Albert of Saxe-Coburg. (The house's name was changed during WWI). The Hanoverians, like several other monarchical houses in Great Britain, were essentially imported monarchs with a) some degree of descent from British royals and b) Protestant sensibilities, who in any case rule with the consent of Parliament.
(Edit: as /u/tim_mcdaniel points out, what I'm getting at here, badly, is that the Act of Succession of 1701 set inheritance standards for a later ancestor of the current Queen, so Richard III's paternal heritage is neither here nor there.)
(To expand on my point 2: it doesn't really matter if Joe the Plumber can prove with DNA that he is a several-generations-descended scion of King Wilifred the Bald of Whateveria, if Whateveria has a constitutionally recognized monarch ruling with the consent of an elected body. In all statistical likelihood, there are tens of thousands of descendants of any given monarch walking around at any given time.)
No implications whatsoever.
The law of succession of the United Kingdom is the Act of Succession 1701 as amended: Electress Sophia of Hanover and the heirs of her body lawfully begotten (with certain Catholic exclusions, marriage permissions, et cetera). It does not depend on any previous succession (except as a movent reason). As well, the "Convention Parliament" of 1689 had been something of a reboot, not really being based on previous law (kind of violating it, in fact), holding the throne ot be vacant, and declaring William and Mary to be king and queen (see The Bill of Rights ).
As I understand the precedent and reasoning behind the "forgery" of the Great Seal on the Care of King During his Illness, etc. Act 1811: for British courts, an act being on the Parliament Rolls with the Great Seal was conclusive proof of it being law and the court could not/would not enquire further, even though it was published that the Chancellor had affixed it without the sign manual of George III. I don't have a source to hand, however.
I've read a mention of law on the subject of presumption of legitimacy, but I don't remember details -- anyone know?
If I can be indulged with a sentence of speculation: given how much governmental chaos would ensue, and given that a large majority accepts the general legitimacy of the current form of government of the UK (albeit some think it ought to be changed in various ways), I cannot believe that this will go anywhere.
None, for the reasons given already. There is also the theory that Queen Victoria wasn't actually the Duke of Cambridge's daughter, that her mother had had a relationship with perhaps a Wittelsbach, resulting in Victoria. This would explain 1) The haemophilia that Victoria was a carrier for, but which the Hanover family did not have. and 2) The Queen's refusal to have a DNA test done.