I'll have to make some rough assumptions to answer this one.
To review, Prince George was born on July 22, 2013 and Princess Charlotte on May 2, 2015. That is a difference of 649 days. If we allow for a nine-month gestation, that is 379 days between the birth of George and the conception of Charlotte.
Just going off of personal observation, let's assume the mean time between siblings is three years. That would mean 825 days between birth and conception of the next child. Using the exponential distribution, I find the probability of a difference of exactly 379 days is 0.0442%.
Next, let's assume that any female between the ages of 20 and 39 is a potential candidate. According to Wikipedia, the population of the United Kingdom in that age range is 16,924,000. Let's divide that by two to get rid of the men for 8,462,000 women of childbearing age in the UK.
The fertility rate in the United Kingdom, which is the average number of children born to each woman of childbearing age, is 1.92. Using the Poisson distribution, I find the probability of two or more children is 69.83%. So, the number of women in the UK of childbearing age who will have two or more children is 8,462,000 × 69.83% = 5,909,015.
Since women generally have children closer to age 20 than 40, let's roughly say that the mother's age at the time of the first born will be evenly distributed between the ages of 20 and 37. So, the number of women who will have their first child in the UK on exactly the birthday of Prince George is 5,909,015/(17×365) = 952.32.
We already established that the probability of an exact age difference between the first and second child of 379 days is 0.0442%. Thus, the expected number of women who had their second child on exactly the same day Princess Charlotte was born, who already had their first born on the exact day Prince George was born, is 952.32 × 0.000442 = 0.421.
Using the exponential distribution, given a mean of 0.421, the probability of at least one woman having her first and second children on the exact same days as Prince George and Princess Charlotte were born is 34.36%.
By the way, I find the probability of the same thing in the United States is 86.32%.