Interesting that some of the crowd did. I did at mine.
Maybe because Liz made little Billy Colonel-in-chief of the Irish Guards right before the do, they didn't have time to get Wilkinson's to rattle him off a brand new sword. In my case, the RSM said, "We haven't got two identical swords that you can borrow and if you are getting married in MY regiment's Peace Time Uniform, you and the best man WILL wear identical cavalry sabres. I have spoken to the QM of the QOH: drive across to Detmold and collect. Return when you get back."
Can you imagine Billy being watched by 2 billion viewers and all the crusty old guards having heart attacks because, "he's wearing a BLIDDY SCOTS Guards sword! Where's the keys to the outrage bus?"
I imagine that as a member of the royal family and as colonel in chief of a regiment, one would enjoy a certain degree of freedom with regard to dress formalities, that being the case, Wills and Harry probably decided that they would rather not limp up the aisle with 3 feet of steel strapped to their sides, especially whilst being scrutinised by about 2 billion people.
After all, modern dress swords are based upon the ones which were designed in the vast majority of cases to be wielded from horseback, making them somewhat clumsy and ungainly when on foot.
A fair assesment as can be seen at the Princess Anne / Mark Phillips wedding...
He could have worn the smaller lighter swords used by Colonels in the Foot Guards unslung, carried in hand, similar to this image of the then Major General Commanding the Household Division, Major General W G Cubitt CB.
The sword, if he does not already have his own, would be furnished by RHQ.
For Prince Harry as a captain in the Blues it would have been similar to Mark Phillips.
My guess is smudge67er was spot on.