Question about the rise of Walpole and office of prime minister

So, his is something I've been pondering both for my current timeline a Britain of panthers abd lions but also for other timelines where the Stuart's are on the throne without a bill of rights.

Would the office of prime minister and the ascendancy of Robert Walpole be inevitable or was it a chance of fate? Could either rise without the glorious revolution of 1688?

Of so, would Walpole necessarily need to be a Whig?
 
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