At my DC's primary school, the christmas play was a huge deal and preparations started early in the term. We had nursery with reception, yr 1 & 2, 3 and 4, 5 and 6. The plays would be run over the course of a day for the parents and all the kids would see the dress rehearsals. Significant amounts of school time in the autumn would be spent rehearsing, learning the songs, lines, costumes, staging etc - obviously with changes in emphasis as they got older.
The plays from year 1 up would always be proper plays with proper scripts and always very much looked forward to by children and parents alike.
BUT, the choice of plays was very varied - retelling of fairy tales, plays with a moral message, plays with a specific Christmas theme. So I've seen the Bossy Christmas Fairy, Santa's on Strike, Handsel and Gretel, Sinderella and many more. I think the choice of play had far more to do with the quality of the scripts, length, number of key parts etc than to what degree it reflected a specifically Christmas theme.
Most school assemblies I attended included a prayer addressed to God and no other deity, even though it is not a church school. So I guess if I'd had strong feelings about it (I didn't), I could have excluded my children. Some people did.
My point is that if I HAD excluded my kids from assemblies I would ABSOLUTELY have expected the school to talk to me about inclusion in the Christmas play rather than just assume they weren't to be involved. I'm not sure the OP said it was specifically a nativity did she? Just that it was the Christmas play - just as ours were always referred to as the Christmas plays regardless of the subject matter.