helen, the bodies in the tower would have a very big impact on the Queen's claim to the throne. Do you remember the show about 'King Michael' about ten years ago claiming that an Australian man was the rightful king of England? The bodies in the tower could confirm or disprove that theory.
Our current royal family's claim goes back to Henry VII and his wife Elizabeth of York via their daughter Margaret. Henry VII was confirmed as King by parliament (so technically any claim via him is valid) but his claim was hugely tenuous and marred both by illegitimacy, acts of parliament disqualifying his line and the fact stronger claimants (including, incidentally, the family of Katherine of Aragon) were passed over to get to him.
The upshot of that is that a lot of people see the legitimacy of the current royal family's claim as coming via his wife Elizabeth of York who was the sister of the Princes in the Tower. But there are questions over their father Edward IV's parentage and rumours that his father was an archer called Blaybourne and not the Duke of York from who his claim came. The date he was conceived is dodgy as his father may have been away at war and he was exceptionally tall and blonde in a small dark family. Richard III was his brother and the princes in the tower his sons. If the DNA of both was examined and it was discovered that the male line was broken and Richard III and Edward IV didn't share the same father it would hugely weaken the current royal family's claim and give a massive boost to republicans.
It's never going to happen much as I wish it would. There are so many mysteries it could solve. There are rumours about Elizabeth I's parentage. We're not sure which bodies in St Peter ad Vincula belong to who, so Anne Boleyn, Katherine Howard, Jane Boleyn and Jane Grey have possibly been mixed up as they were only identified by poor Victorian techniques. And rumours of poisoning of people such as King John could be investigated.
You never know, maybe Charles will change it, but I'm not holding my breath.