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I think this should happen to several families per year, because of the chances of people sharing a birthday.
So, the first baby to be born can have their birthday on any day of the year, so there is a chance of 1 that it is the right day, because it doesn’t matter when it is. The second baby has a chance of 1/365 of being born on the same day as the first. Then, the chance of the third baby being born on the same day is 1/365. So, you have to multiply the fractions together to work out the chance of them all being born on the same day. 1x 1/365 x1/365= 1/33225.
So, if there are a million families in the UK with 3 children then the chances say that roughly 31 of them have a shared birthday. However, this could be completely inaccurate, as it is just chance. There could be 999,999 families like this or just 1 family.