My birthday is Christmas Day and I've never really had an issue with it.
I know a few people with Christmas Birthdays who had summer parties as kids (like the Queen, lol) but as my little brother's birthday is June 26th, that would've taken away from him.
I love that I've only worked one birthday in my life and that I get to spend the day with family and it seems like the whole country is decorated for my birthday! When I was little, my Mum always made sure we had birthday cake on Christmas Day as well. My Mum was also really good about getting non-Christmas wrapping paper for my birthday presents - and making sure that I didn't get too many joint presents (it's great if you want something big though!).
Ironically, I'm now pregnant and due Dec 19. My Mum reminded me that I was due on Dec 14, so I may still have a Christmas Day baby!
If your baby is due Dec 26th, I wouldn't plan to be going anywhere at Christmas or making much plans at all. Your baby will be full term at anything from Dec 5th to Jan 9th, so I really wouldn't be travelling then.
I think we'll be at home, stocked up for having Christmas dinner just the two of us (or hopefully 3!). What we've done some years anyway is have the family round for one day between Christmas and New Years, so if pip arrives on time, we might do a meet-the-baby day here. I saw a suggestion elsewhere of having an "open house" day a few days after baby is born, so everyone can come and visit, but then needs to go away again and give you some peace. It sounded quite a good idea.