For birthday pass does it have to be the exact date or can we go the next day?

Yes, it has to be their actual birthday or the closest day the park is open to their actual birthday

My nephew's b-day is this month, he'll be 5 he wants to go to Lego Land. Does he gets a free entry then? .

During the month of their Birthday. I was there on Wednesday and I asked for my daughter.

Birthday

It has to be either the day of their actual birthday or the closest day the park is open to their actual birthday.