Very very hard. In the first place, 'Er Maj was not born till 1926, long after the German monarchy was abolished. So the PoD has to revive the German monarchy after 1926, which is very hard. (And no, butterflies cannot be avoided - it must be "our own Queen Lizzy", not some analog with the same name.)
Next, one has to have that throne come to her - and she is not even remotely descended from the Hohenzollerns.
However - I'll take one stab at it: there are numerous failures of republican democracy, empowering more fascist and Communist dictators. In some cases, monarchy either thwarts the dictatorship or is destroyed fighting it. (Things Happen to reverse the OTL bad examples of Italy and Spain.) The dictatorships cause a general war in Europe, and at the end of it, Britain with some minor allies has defeated all the Bad Guys.
The war ends around 1960. Elizabeth has been a heroic not-just-symbolic war leader for Britain; she personally resolved parliamentary crises, and at least once insured the selection as PM of a fighter instead of a surrenderer. She was seriously wounded in an air raid, and lost one leg because she insisted that some injured children and soldiers be treated ahead of her; at the end of the war she raised great sums to provide for the orphans and cripples left by the war throughout Europe.
There is a general consensus that without a monarch, democracy is unreliable. Also, to prevent future wars, Europe becomes a federal union. Elizabeth is made head of state, using the title "Empress". Those countries which have no monarchy of their own nominate Elizabeth as their local monarch, including Germany.