1. Which British Royal? And why would they be there?
2. What could be the future effects of this? Russian Pride in the Balkans could be destroyed by this.
1. There's flexibility. One of George V's sons on an Adriatic cruise maybe, but anyone in the extended Royal family would do. The visit wouldn't be an official one -- it would be enough if the visitor was on good personal terms with Franz Ferdinand and was invited by him to ride in the procession then go hunting afterwards. If the British Foreign Office had cautioned the prince in advance not to associate with Franz Ferdinand, it wouldn't happen, but I'm not sure the F.O was sharp enough to do that, and the princes weren't so astute to predict any major ramifications themselves.
On your earlier point, I don't think that an unofficial Royal visit to a Central Power would be too unreasonable. The Russian Royal family visited with the Kaiser before the war, and royal visits were sometimes a deliberate prelude to more formal diplomacy. Edward VII's Royal visit to France is a good example. If needed, additional facts can be butterflied in -- the prince's yacht is wrecked in the Adriatic and he and his party are rescued by an Austrian warship for instance.
2. Effect: having thrown in a monkey wrench, I was going to let everyone else chime in on the consequences. The obvious one is that I couldn't see Britain allying with Serbia after the assassination, no matter what the treaties required.