I imagine Network Rail do their damnedest to ensure the Royal Train always gets through. Signallers are made aware that the royal train is in their neck of the woods as it carries a unique headcode. 1X-- I believe. Can't be seen to unnecessarily delay Brenda. One would not be amused.
Perhaps her Maj should take more journeys on normal passenger workings to get a flavour of what her subjects have to often put up with.
She has slummed it on occasions:Link
to Reuters article.
Although I believe the proles were turfed out of this particular 1st Class section of a FCC train!