When my son turned one we already knew about the milk and egg allergies and my husband and I decided that his birthday would be a completely safe birthday for him. We didn't want to have any food at the party that could pose a threat to his health and safety. We baked a cake for him from scratch and bought a few extra foods and drinks to serve. However, we also decided to make sure the party wasn't food focused. Thus, we went to the zoo.
There were a few moments of pure comedy though the night before. After I baked the applesauce cake that was 100% safe I put it to cool on the table. My nephew who was visiting us and is 2 years old was eating a yogurt and getting messy as all 2 year old children do when they eat. Not really a problem considering he wasn't eating over the cake. However, when he was done he saw the cake took his dirty yogurt covered hands and went straight for the cake smearing the top with two lovely hand prints. Take two of the cake worked much better and I hid it from little hands.
Other than that the birthday worked really well and my son had a special first birthday that was just for him.