We have celebrated my sons birthday there each year. For the first birthday I would try and keep it simple. Maybe even a birthday lunch since it will be cheaper with just as good of opitions. The Crystal Palace in MK is great for the little ones. Pooh and all his friends are there. They do a birthday dance and all the birthday goers get mickey ears, and fun stuff at the table. Just make sure when you make the reservations you let them know. They will also have all the charaters sign a card for your child. A cute keepsake.
Another fun place is Chef Mickeys-even for Breakfast. What better way to celebrate a little ones birthday then in the AM when they are usually the most alert and happy. They have a fun dance and great interacting with the kids too.
Good luck and have fun!
-Also most of the resort resaurtants will bring out a big cupcake with a candle in it for a birthday. So for a 1 year old that is plenty. If you wanted a cake for the family special ordering one is a good idea or the Boardwalk bakery usually has some nice cakes too. Dont forget about the Birthday Button either. You can get them at the front desk of your resort or at any park.