M.K.
My son will be celebrating his first birthday near Christmas too. I'm not doing a Christmas theme for him, but I think it could be cute for a first birthday. After that I'd try to keep it separate from Christmas so they do feel like they are getting a real birthday celebration and not just an extra Christmas party with their name on a cake.
I think keeping it simple with just close family is best for a first birthday. For later birthdays a friend once told me to invite their age plus one in friends.