Great Britain has an unwritten, evolving constitution based on law and precedent.  Were the Royals to obviously attempt to build a political base and to intervene in elective politics that would overturn a century of law and precedent and would likely constitute political suicide for The Monarchy.

As the Dutch said while fighting the Spanish: "It is not necessary to have hope in order to persevere."

They have a base, in the uniquely British flock of both minor and major legacy aristos, and some understated but far from irrelevant links to Parliament. Phil and Brenda can't propose legislation, but they can hint that actions that adversely affect their interests may be unwelcome. And as the aristos have been finding it hard to keep the old pile repaired, they've been rubbing shoulders - and allegedly other body parts too - with the cream of the old-Etonian business world.

So - not primary players, but they have vast piles of cash, property and contacts, which shouldn't be underestimated, and which make anything as vulgar as a party machine somewhat redundant.